The Asymptotic Lower $r$-Permanent Conjecture

Let $r\geq2$, and let $B_n$, $n=1,2,\ldots$, be a sequence of $n\times n$ doubly stochastic matrices such that every column of every $B_n$ has at most $r$ nonzero entries. Let $k_n\in[0,n]$ be integers such that

$$\lim_{n\to\infty}\frac{k_n}{n}=p\in(0,1].$$

For a nonnegative matrix $B$, let $\operatorname{perm}_{k}B$ denote its $k$-permanent, and let $gh_r(p)$ be the function defined in the Asymptotic Lower Matching Conjecture.

The Asymptotic Lower $r$-Permanent Conjecture. Under these hypotheses,

$$\liminf_{n\to\infty}\frac{\log\operatorname{perm}_{k_n}B_n}{2n}\geq gh_r(p)-\frac{p}{2}\log r.$$

This is presented as a stronger conjecture implying the Asymptotic Lower Matching Conjecture, by applying it to normalized incidence matrices of regular bipartite graphs. Its general validity is left open.