The recursive-construction conjecture for permanents of 312-avoiding matrices

Let $f_{312}(n)$ be the maximum permanent of an $n\times n$ 0-1 matrix avoiding the permutation pattern 312. Let $\mathcal{M}_n(\ast,\circ)$ be the set of $n\times n$ matrices obtained by starting with $1\times1$ identity matrices and repeatedly applying the two operations $\ast$ and $\circ$ in any order. Recursive-construction conjecture. For every $n\geq1$,

$$f_{312}(n)=\max\{\operatorname{per}(A):A\in\mathcal{M}_n(\ast,\circ)\}.$$

The conjecture concerns whether the displayed recursive operations always contain an extremal construction. The source states it as open and does not provide a proof or counterexample.