Lower-bound conjecture for spanning trees containing a forest in complete tripartite graphs

Let $X_1,X_2$ and $X_3$ be the partite sets of the complete tripartite graph $K_{n_1,n_2,n_3}$, where $|X_i|=n_i$ for $i\in \llbracket 3\rrbracket$. Let $F$ be a spanning forest in $K_{n_1,n_2,n_3}$ with $k$ components $T_1,T_2,\ldots,T_k$. Write $n=n_1+n_2+n_3$ and $n_{s,i}=|X_s\cap V(T_i)|$ for $s=1,2,3$ and $i\in \llbracket k\rrbracket$. The lower-bound conjecture. The number $\tau_F(K_{n_1,n_2,n_3})$ of spanning trees of $K_{n_1,n_2,n_3}$ containing all edges of $F$ satisfies

$$\tau_F(K_{n_1,n_2,n_3})\geq \frac{1}{n_1n_2+n_1n_3+n_2n_3}\left(\prod_{i=1}^{k}\left((n-n_1)n_{1,i}+(n-n_2)n_{2,i}+(n-n_3)n_{3,i}\right)\right)$$ $$\qquad\cdot\left(1-\sum_{i=1}^{k}\frac{n_{1,i}n_{2,i}+n_{1,i}n_{3,i}+n_{2,i}n_{3,i}}{(n-n_1)n_{1,i}+(n-n_2)n_{2,i}+(n-n_3)n_{3,i}}\right).$$

This conjecture seeks a general lower bound for the number of spanning trees containing a prescribed spanning forest in a complete tripartite graph; its status is not resolved in the supplied source material.