Despite Israeli Prime Minister Benjamin Netanyahu comprehensively rejecting Hamas's ceasefire counter-proposal, Blinken said it "creates space for an agreement to be reached". He pledged the US would continue to "work relentlessly" to achieve a ceasefire and hostage release deal.

Putting aside the fact the job description of senior diplomats requires them to remain upbeat in the face of negotiating setbacks, does Blinken's shuttle diplomacy - he has visited the Middle East five times since Hamas's October 7 attacks on Israel - have any chance of success?

Israel and the US presented a proposal to Hamas via Qatar about a week ago. It was not made public, but Qatar's Al Jazeera news agency reported sources "close to the talks" as saying it involved an initial 40-day truce, during which Hamas would free the remaining Israeli civilian hostages it holds, followed by Israeli soldiers and the remains of dead hostages.

Hamas's counter-proposal, delivered on February 7, offered freedom for all remaining hostages and the return of the deceased in a three-stage ceasefire lasting 4.5 months. In return, Israel would first release all Palestinian women and children held in Israeli jails, as well as 1500 male prisoners, including 500 serving long sentences.

At the same time, the Israeli military would implement a phased withdrawal of its troops from Gaza, and the ceasefire would become permanent. The obvious implication of the proposal was that Hamas would remain in control of Gaza.

It's not surprising each set of proposals was unacceptable to the other party. Israel didn't offer any guarantees it wouldn't resume its military campaign after the release of the hostages. And Hamas's proposal was effectively a return to the status quo before October 7, which would be entirely unacceptable to the Netanyahu government. Each proposal appeared to represent the maximalist positions of each side. As such, the standard technique of practised negotiators is to examine both proposals and look for - or try to create - common ground for a deal. Can that work now?

Despite Netanyahu's stern rebuff of Hamas's counter-proposal, a Hamas delegation has travelled to Cairo this week for more ceasefire talks. But whether Netanyahu is prepared to keep talking will depend on his evaluation of the pressures he faces on three fronts:

Netanyahu is beholden to prominent hardliners in his right-wing government, particularly Finance Minister Bezalel Smotrich and National Security Minister Itamar Ben-Gvir. Ben-Gvir threatened to bring down his government over any attempt to enter a "reckless" deal with Hamas to free the hostages. If Netanyahu is forced to hold new elections, opinion polls show he would have very little chance of forming a new administration.

Second, the families of the 136 hostages still held in Gaza and their supporters hold daily demonstrations demanding the government prioritise negotiating their release over the military campaign against Hamas. The news that 31 of the hostages have been confirmed dead can be expected to raise the families' anxiety levels and increase the tempo of their protests.

And third, Netanyahu faces increasing pressure from the Biden administration, which is suffering reputational damage across the Middle East and in the Global South because of its unconditional support (including providing weapons) for Israel's Gaza campaign. Within the US, Biden is also experiencing blow-back from young, progressive Democrats, horrified at the Palestinian death toll, which now stands at over 27,000. That could affect his re-election prospects if they decide not to turn out for him in the November vote.

By comparison, the pressures on Hamas are of a lower order. Obviously, Israel's military campaign, particularly its current extension into southern Gaza, is causing enormous suffering to the civilian population. But the degree to which this affects the Hamas leadership is uncertain.

In negotiating through Qatar and Egypt for a ceasefire, an increase in aid and, ultimately, an end to the conflict, Hamas is presumably motivated - at least partly - by a desire to reduce civilian suffering. But its main aim is unquestionably its own survival. What would force Hamas to compromise on its demands would be the capture or deaths of its senior leaders, Yahya Sinwar and Mohammed Deif.

It should be noted Hamas derives some benefits from the continuing conflict. What Biden has described as Israel's "indiscriminate bombing" campaign actually boosts Hamas's image as a standard bearer for Palestinian rights.

The Gaza war, with its horrifying human toll, has brought the Palestinians' plight to international attention and harmed Israel's global standing.

Hamas would also be aware that it does not have to defeat Israel militarily in order to win this war. It needs merely to survive. A ceasefire that left Hamas in charge of a Gaza in ruins would thus be a victory.

Unless there is an unexpected development - Israel's elimination of Sinwar and Deif, or its military locating and freeing remaining hostages - the war is likely to continue for some months.

Netanyahu probably feels he has no choice, from a political perspective, but to continue prosecuting the war in the same manner, in the hope of a breakthrough. His history of staring down US presidents means he almost certainly won't back down under pressure from Biden. And he will continue to tell the hostages' families that their loved ones can only be rescued by military action alone.

Netanyahu may be prepared to sanction renewed temporary ceasefire offers to Hamas in an effort to win more hostage releases - but not if doing so puts his governing coalition at risk.

The entrenched positions of the Netanyahu government and the Hamas leadership mean Blinken's work is nowhere close to being done. That means more trips to the region, more shuttle diplomacy and, likely, more sleepless nights.