The permanent bound for 1-factors of simple 3-uniform hypergraphs

Let $G$ be a simple $3$-uniform hypergraph, let $M(G)$ be its adjacency matrix, let $\operatorname{per} M(G)$ denote the permanent of that matrix, and let $\varphi(G)$ denote the number of 1-factors of $G$. Permanent bound conjecture. The number of 1-factors satisfies

$$\varphi(G)\leq \left(\operatorname{per} M(G)\right)^{1/3}.$$

The analogous bound is established in the surrounding results for uniformities other than $3$, but the authors state that they were unable to prove this exceptional case and believe it is likely to be true.