I think the thing that would bring relegation worries quickest right now is the unknown if we got rid of Sheehan. Today was bad, but in every other game we are competitive and we look like a midtable side who will win some and lose some and draw some and survive comfortably.
Unless we go on a run of multiple successive defeats, he should remain for a while yet. Even if the club internally decide to replace him but let him stay in charge for the short term until the right replacement is there. The club seem to like their data nowadays and I am sure it will not be beyond them that we just do not look like a side capable of creating many chances or scoring many goals and they will know it is hard to consistently win without that. I am slightly sceptical, but if Sheehan can add that at some point consistently in the next couple of months then it will be a positive step. If we are still struggling to create multiple chances a game from open play after that, then maybe it is time to prepare for a new manager even if it does not actually happen until next season.
That may sound accepting of mediocrity but these things need time to truly see what is working and what is not.