Amdeberhan–Moll divisibility conjecture for binomial coefficients

Let $a,b$ and $n$ be positive integers with $a>b$. For integers $x$ and $y$, write $x\mid y$ when $x$ divides $y$. Amdeberhan–Moll's divisibility conjecture.

$$(2bn+1)(2bn+3)\binom{2bn}{bn}\mid 3(a-b)(3a-b)\binom{2an}{an}\binom{an}{bn}.$$

The conjecture concerns divisibility properties of products of binomial coefficients. It is stated here as a conjecture of T. Amdeberhan and V. H. Moll and is proved in the paper, so its status is solved.