The permanent moment growth conjecture with fixed matrix size

Let $M$ be a random $k\times k$ matrix with independent standard complex Gaussian entries, where $k\geq 3$ is an integer, and let $t\geq 3$ be real. The non-integer moment growth conjecture. There is a quantity $C_{k,t}$ such that

$$\mathbb E\left|\operatorname{Perm}(M)\right|^{2t}=C_{k,t}\frac{k!^{2t}t!^{2k}}{(kt)!},$$

and

$$\lim_{k\to\infty}\frac{\log C_{k,t}}{k}=0.$$

This is the real-moment form assumed later to derive concentration results from the moment-growth heuristic. The source gives no proof or resolution.