Pfaffian conjecture for the vertically symmetric alternating-sign-matrix sequence

Let $a_n=\frac{1}{2n+1}\binom{3n}{n}=\frac{1}{3n+1}\binom{3n+1}{n}$. Vertically symmetric alternating-sign-matrix Pfaffian conjecture. For every integer $n\geq1$,

$$\mathop{\rm Pf}\left((j-i)a_{i+j-1}\right)_{1\leq i,j\leq2n}=\frac{1}{2^n}\prod_{k=1}^{n}\frac{(12k-6)!(4k-3)!(3k-1)!}{(8k-6)!(8k-3)!(3k-2)!}.$$

The sequence $a_n$ also occurs in determinant enumerations of vertically symmetric alternating sign matrices. The displayed Pfaffian identity is proposed on the basis of that connection and remains open.