Amdeberhan's divisibility conjecture for binomial coefficients

Let $a,b$ and $n$ be positive integers with $a>b$. Amdeberhan's divisibility conjecture.

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

This is a generalization of a theorem in the paper, found by T. Amdeberhan in personal communication. The text says that a proof would be interesting and gives no resolution, so the conjecture remains open.