Nick Hunter-Jones's moment conjecture for permanents of Haar unitary matrices

Let $U(d)$ denote a Haar-random $d\times d$ unitary matrix, and let $t$ be a positive integer. Hunter-Jones's conjecture. The permanent moment satisfies

$$\mathbb E_{M\sim U(d)}\left|\operatorname{Perm}(M)\right|^{2t}\approx \frac{t!}{\binom{2d-1}{d}^{t}}.$$

The source attributes this conjecture to Nick Hunter-Jones and supports it with numerical experiments. It is compared with the lower bound for moments of permanents of Haar-random unitary minors, but no resolution is given.