I admit that the "best case scenario" is wildly optimistic. Wildly. And I do not expect that it will happen. But it is "possible." It projects wins against West Brom (h), Wigan (a), Southampton (a), Villa (a), West Ham (h), Reading (a), Everton (h), Fulham (a), and QPR (h); draws against Swansea (h), Tottenham (h), and Newcastle (a); and a loss to Chelsea (h). The three matches against sides above Liverpool in the table are all at Anfield, the six away matches are all against sides currently placed between 13th and 19th. Regardless, I will be incredibly surprised if Liverpool managed 9W-3D-1L during this stretch, no matter the opposition.
Liverpool's points per game over the first 25 matches is a meager 1.44, better than last season's total (1.37) or the dismal 1.25 earned under Hodgson, but still nowhere good enough. Were Liverpool to finish the season with the same points-per-game average, they'd end on 55 points. Since the league switched to three points for a win, Liverpool have finished with a worse total just twice: 54 points in 1998-99 (Houllier's first season) and 52 in last year's campaign.
However, if you ignore the opening five matches of the season, where Liverpool took just two points, the points-per-game average rises to 1.70. Which, incidentally, is also Liverpool's points per game average in the last ten matches. Were Liverpool to continue that form, they'd finish with 58 points – the same number as in Benitez's first season, as well as 2010-11.
Liverpool have bettered that points-per-game average in the last six matches, despite traveling to the 1st, 2nd, and 6th-placed sides. By averaging 1.83 points per game for the rest of the season, Liverpool would finish with 60 points. Which actually borders on respectable. But 60 points has gotten fourth place just once in the last ten years: 2003-04. That was actually Liverpool's total in Houllier's final season. But chances are, 60 points won't do it this season; Tottenham already have 45, Everton 42, and Arsenal 41.
And if you go by Liverpool's results in equivalent fixtures last season, they'd reach 56 points: wins over Southampton (replacing Blackburn), Villa, West Ham (replacing Wolves), Chelsea, Everton, and QPR; draws against Swansea and Wigan; and losses to West Brom, Tottenham, Reading (replacing Bolton), Newcastle, and Fulham.
If you do the same for the four teams directly above Liverpool, Arsenal and Everton actually have worse comparable results in last season's fixtures. Arsenal won five, drew one, and lost seven; Everton won three, drew three, and lost seven. That'd give Arsenal 57 points and Everton 54 points, both catchable even with mediocre Liverpool results over the next 13 matches. But both Tottenham and Chelsea decimated their comparable fixtures last season – Chelsea's record was 8W-2D-3L, Tottenham's 9W-2D-2L, which would give Chelsea 72 points and Tottenham 74.
Over the last 10 seasons, the average number of points needed for fourth place is 67.8. Tottenham would need 1.77 points per game – say, 6W-5D-2L or 7W-2D-4L – to reach that mark. Liverpool would need 10W-2D-1L, two points more than the above "best case scenario."
Liverpool aren't reaching 68 points. And Liverpool probably won't reach 4th, no matter how many interviews with players they post on the official site. But Liverpool will almost certainly better last season's point total, while playing better football. And, honestly, after the last few seasons, discernible, continued progress remains all we can really hope for.