Despite leading Wednesday to back-to-back top-six finishes, Carvalhal is rapidly losing backers as the Owls struggle to keep pace with the play-off pack this campaign. His chairman, Dejphon Chansiri, however, seems so far unmoved.
Surely, though, there will come a point at which Chansiri has little choice but to bow to fan pressure - barring an almost unthinkable improvement in fortunes and, for many Owls followers, performances too.
The Thai may, however, be wary of what happens after Carvalhal, with any new boss unlikely to see the funds the Portuguese has enjoyed in his two and a half years at Hillsborough - even if he is not entirely responsible for how some of that was spent.
Someone who's delivered on a budget, then, would be a decent place to start. But there are other criteria to consider, too. Does he know the Championship? Would he work within a recruitment structure such as Wednesday's (which Chansiri admits is made up of numerous external advisers)? Could he spot a player if required to himself? Would his brand of football be deemed desirable by an expectant Owls fanbase?
Brentford's Dean Smith, arguably, ticks all of those boxes. Let's consider too how the likes of Jota, Scott Hogan and, currently, Ollie Watkins, have significantly bettered themselves under Smith's management - an area in which Carvalhal is often perceived to fall short.
Okay, so he's yet to win anything in six years as a manager, but how many would at Walsall and Brentford - two clubs at which Smith has often found his best players swept from under him?
And if Wednesday are going to tempt any working boss to swap his role for one at Hillsborough, their former captain - albeit not the most successful or abiding one - could be the one to take the bait.
Would you want Smith as Sheffield Wednesday's next manager?