Three-dimensional Catalan formula for reduced valid hook configurations

For $k\geq 1$, let $\operatorname{Av}_n(312)$ denote the set of permutations of length $n$ avoiding the pattern $312$, and let $\operatorname{RedVHC}_k(\operatorname{Av}_n(312))$ be the reduced valid hook configurations with $k$ hooks on such permutations. Then

Three-dimensional Catalan conjecture. For every $k\geq 1$,

$$\left|\operatorname{RedVHC}_k(\operatorname{Av}_{3k}(312))\right|=2\frac{(3k)!}{k!(k+1)!(k+2)!}.$$

The right-hand side is known as a three-dimensional Catalan number and appears in OEIS A005789. The source presents this as a data-motivated conjecture and supplies no evidence of resolution.