I don't think they'll be under any illusions about how pish we are, but it's been 3 months of pish. In his first 12 months he had us 3rd in the calendar year - there's the proof that he does know how to win games. He's got plenty left in the bank from those first 12 months, although I imagine it's rapidly dwindling. We're also only probably a couple of wins away from an "okay" season (i.e. finishing 6th / 7th), and still in touching distance of Europe, and the board have to consider the financial implications of ditching a manger with a couple of years left on his contract - won't be cheap at all. Aye, we're in relegation form but it's highly unlikely we'll be relegated. If the board stick or twist for the board and it's a gamble either way.
I think the turning point will be when the players have switched off to the message. As soon as they've turned on the manager, he's fucked. If that's not happened yet then GA will be here for the foreseeable future. Hate to say this I can picture us scraping through to the end of the season with him, finishing 9th or something and GA getting the league cup groups and first month or so of next season to prove this summers signings were the answer he needed.