Consider the sequence of [Apéry numbers][1]  
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{k}{n}^2\binom{n+k}{k}^2 .
$$
In an email, physicist [Alan Sokal][2] conjectures that it is a [Stieltjes moment sequence][3].  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 http://mathoverflow.net/questions/178790/ , where a formula for $A_n$ was established.]


  [1]: http://oeis.org/A005259
  [2]: http://www.physics.nyu.edu/faculty/sokal/
  [3]: http://en.wikipedia.org/wiki/Stieltjes_moment_problem