The extremal matching conjecture for almost alternating threshold graphs

Let $k\geq 0$. Let $A$ be an almost alternating threshold graph, and let $G$ be a threshold graph with the same number of vertices and edges as $A$. Write $m_k(H)$ for the number of matchings of size $k$ in a graph $H$. Extremal matching conjecture. Then

$$m_k(G)\leq m_k(A).$$

In addition, if $k\geq 2$, $m_k(A)>0$, and $G$ is not almost alternating, then

$$m_k(G)<m_k(A).$$

The paper establishes that almost alternating threshold graphs maximize the total number of matchings, but its methods do not extend to matchings of a fixed size $k$. Computer testing through $22$ vertices supports this stronger fixed-size assertion, which remains open.