Product formula for successive vertex orderings of the line graph of a bipartite 3-uniform hypergraph

Let $K_{m,n}^{(1,2)}$ be the $3$-uniform hypergraph on $V=V_1\cup V_2$, with $|V_1|=m$ and $|V_2|=n$, whose edges satisfy $|e\cap V_1|=1$ and $|e\cap V_2|=2$. Let $L(K_{m,n}^{(1,2)})$ be its line graph, and let $a_i=(m-i)\binom{n-2i}{2}$. Define

$$d_i=\frac{a_0-a_i}{i}.$$

Product-formula conjecture. The number of successive vertex orderings satisfies

$$\sigma'(L(K_{m,n}^{(1,2)}))=m\cdot\prod_{i=1}^{m-1}\frac{mn-\binom{m+1}{2}+\binom{i}{2}}{d_i},$$

where each fraction is evaluated after disregarding all zero factors in both the numerator and denominator.

The paper presents this as a product formula complementing an earlier summation formula. Its status is not resolved in the supplied text.