Unique-minimal-element conjecture for ribbons with two adjacent row lengths

Let $\mathcal{R}((a+1)^n a^m)$ be the poset of ribbons whose row lengths consist of $n$ copies of $a+1$ and $m$ copies of $a$, ordered by Schur-positivity. Unique-minimal-element conjecture. In the poset $\mathcal{R}((a+1)^n a^m)$ there is exactly one minimal element, given by

$$(a+1)^{\left\lceil\frac{n}{2}\right\rceil}a^m(a+1)^{\left\lfloor\frac{n}{2}\right\rfloor}.$$

This conjecture concerns the opposite extremal problem from the maximal-element conjecture and is attributed in the source to McNamara and Pylyavskyy. The supplied text gives no resolution, so its general status remains open.