When you say "eventually"... how long is that?
Keep in mind that if it's longer than four or five seconds... you're probably fine.
(Can you think of a flight condition where you'd be dead vertical, nose up, at idle, for longer than that?)

That said... you might make sure the muffler to crank seal is good, and that there are no
leaks in the pressure line. Still... I have to believe that most engines at idle will eventually
sag lean and die, if the plane is held completely vertical for any length of time.
There's just not enough pressure to keep forcing the fuel up to the carb.

I could be wrong, of course... never tried it myself

<edit- typo>