Unique-maximal-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. Let $P_{R,S}$ denote the box diagonal diagram with $R$ rows and $S$ columns. Unique-maximal-element conjecture. In the poset $\mathcal{R}((a+1)^n a^m)$ there is exactly one maximal element, given by

$$P_{n+m,(n+m)a-m+1}.$$

The conjecture is a specialization of the proposed maximal-elements description for connected skew shapes. The paper proves it in certain cases, including appropriate chains and short-end cases, and confirms uniqueness in an even case, but not in full generality.