A section of your own fans are calling for his head. He's done a good job taking you back up into top 4 contention after Puel, but I think the psychological effect of bottling it two years in a row could be too much for some people. Winning the cup could buy him another year or two, but I wouldn't be surprised if he suddenly resigns by mutual consent and ends up at Spurs if he doesn't.
Unfortunately, you'll always get the types of characters, especially in these days of instant gratification, who get spoiled by success and expect that to become the norm, or who simply aren't capable of seeing the bigger picture, or of rational thought. It's a very small minority of fans on the forums who'd want him out.
Rodgers has clearly done a really really good job. To be in an FA cup final and Top 4 contention again, especially considering the considerable injuries we've had this season, is an excellent achievement. There isn't much upside to losing him, whilst there is a lot of downside. He's about the best manager a club of our size could attract, although I'd also say we've been good for him too given the young talented squad he inherited, and with owners genuinely invested in the success and continued growth of the club.
I'd be reluctant to automatically say we'd have bottled it if we don't get Top 4 this season, although clearly the media narrative will suggest otherwise, and the disappointment amongst our own fans would be massive given the circumstances. Newcastle debacle aside, only 2 teams have won more points than us over the last 5 games and we have 2 difficult fixtures to finish, which we could easily lose on any given day with no-one really batting an eyelid. We could conceivably pick 3 points up from those 2 games and still not qualify for the CL, and it'd be hard to really suggest a bottling if that happens given Liverpool would have picked up 23 points from 9 games if that scenario played out.
If we don't get top 4, i'd be looking at meek home defeats to Fulham, West Ham, Leeds, Newcastle, Arsenal and even Everton first, before looking to point the finger at the squad and manager mentality at the business end of the season.