Mark, yes, the positivity of this infinite set of finite sums feels to me quite similar to (and, as you point out, implies) the positivity of the infinite series I asked about earlier in the question you reference, and suffers from the same delicate cancellation of huge quantities. I sort of suspect that if you could crack the infinite series, you could crack this, too.

Gottfried, thank you for the numerical evaluation. This is consistent with how I believe the series behaves and with the numerical calculations I have done, although I didn't have the tools to push it out anywhere near as far as you did. A comment and an open-ended question: Presumably one could use interval arithmetic to produce a computer-aided proof of positivity for some negative values of $x$. Suppose you could prove positivity down to some large negative value. How large in magnitude would that $x$ have to be for you to really believe, absent a proof, that the series is positive?

@Gottfried: Thank you for pointing out the related math.stackexchange question (looking at $(n!)^\alpha$ for $\alpha \approx 2 > 1$. I did not realize that there was a connection to Bessel functions.

@Mark: Yes, I don't have any reason to believe that $\sqrt{n!}$ is any different than $(n!)^\alpha$, where $0 < \alpha < 1$ (although, hypothetically, $\alpha = {1 \over 2}$ might be an easier special case).

@Igor: I don't have any compelling evidence that this conjecture is true (although I believe it is). My best evidence is straightforward numerical experimentation. Also, intuition leads me to think it's true, and when I get stuck trying to prove it it's because "this intermediate step doesn't seem to help much," rather than "this intermediate step seems not to be true."