Bryan Jackson and Trevor Birch of administrators BDO are in Lithuania trying to persuade the creditors' committees of UBIG and UKIO Bankas, two parts of former owner Vladimir Romanov's collapsed banking empire - both of which are in administration themselves - to accept a £2.5 million settlement so that the club can move on via a company voluntary arrangement (CVA).
It is a complex, shifting situation, but if they are successful the club can aspire to a bright, shiny future, albeit in the SPFL Championship, under the ownership of Ann Budge and, eventually, the Foundation of Hearts (FOH).
If no deal is possible - Hearts need the assent of both creditors, and as yet it remains unclear whether even a meeting at UKIO will go ahead - the money will run out at the end of April and all bets are off. Although Budge has not indicated that she would back an alternative, albeit rather unpalatable, move to restart the club post-liquidation on the Rangers model, most probably from the bottom tier, such a scenario is at least a plausible back-up plan. Ironically, though, the confirmation of their relegation from the top tier for the first time in 31 years, and consequent drop in the club's share value, may help, rather than hinder, the administrator's case of a CVA.
News will come one way or another by mid-to-late afternoon today and it promises to be a fraught business for everyone associated with the club, not least the players who should be enjoying a day off. "I believe we'll get a call or a message on Monday," said midfielder Scott Robinson. "I'm sure the decision will be out there. I'm not going to sit glued to the TV, though. I'll try to keep my mind off things, hope for good news. I might just take it easy, meet up with some mates, go for a game of snooker or pool. Everyone at the club would be devastated if it was bad news, after thinking it would be positive for so long."
The way Hearts succumbed to relegation was somehow emblematic of their entire season - a rag-tag group of enthusiastic young players, chaperoned by a couple of older heads, bravely taking on the odds by winning in style at Firhill, only to be condemned by events unfolding elsewhere, namely the two late goals by which St Mirren beat Motherwell. For Robinson, thoughts were already stretching ahead to next season, and how these young players, with a year's extra maturity behind them, can bounce straight back to the top flight - from a division which will also contain Rangers. He was happy to add his name to a list including Ryan Stevenson of players content to stay with the club, wherever they end up.
"I've got another year on my deal, plus an option after that," said Robinson. "I'm definitely keen to stay. This is the club I want to play for, no matter what division we're in. We've gone down but hopefully we come straight back up. I think most of the boys would say the same. I've not heard anyone say that just because we were getting relegated, they wanted to leave. They will know themselves, deep inside, but everyone seems keen. We're fighting for the cause, fighting for one another and we've improved as a team together. It's just unfortunate we couldn't get a few wins earlier in the season."
"It would be great going to Ibrox again next year, Rangers coming back to Tynecastle as well," he added. "If we can get a good start and get to the top of the league, hopefully it will be us two fighting it out for the automatic promotion spot. And we could have Hibs down there, too. I've nothing specific planned for Monday but I'll certainly watch the Hibs game at night - and be willing Aberdeen on."
So too will Partick Thistle. Because after a bright, inventive start, this match soon became every bit as traumatic for them as it was for Hearts. It didn't seem that way when Kris Doolan was giving the hosts a thoroughly deserved early lead, but goals from Dale Carrick, Billy King and a Stevenson double, despite Jordan McMillan's late consolation, left them rooted to second bottom and facing a relegation play-off.
"There's four teams and four points separating us all and there's 15 points to play for, so nobody is safe," said Partick Thistle centre-half Lee Mair, who finds himself vying with his former team to finish above second bottom. "I have a little bit of banter with the St Mirren boys but it's livelihoods you're talking about and you don't want to see anyone go down - it's horrible."