Consider the sequence of Apéry numbers
$$ A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3 = \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 . $$ In an email, physicist Alan Sokal conjectures that it is a Stieltjes moment sequence. That is, that there exists a probability measure $\mu$ on $[0,+\infty)$ so that $$ A_n = \int_{[0,+\infty)} s^n\;d\mu(s) \tag{1}$$ for $n = 0,1,2,\dots$. [Of course you can equivalently say that $\mu$ is a nondecreasing function with $\mu(0)=0$ and $\lim_{x\to+\infty} \mu(s) = 1$ and that (1) is a Stieltjes integral, rather than a "measure" integral.]

Is that conjecture correct? Is $A_n$ a Stieltjes moment sequence?

[This question is a follow-up to https://mathoverflow.net/questions/178790/ , where a formula for $A_n$ was established.]