Engbers–Erey's extremal conjecture for average maximal matching size in trees

Let $T$ be a tree of order $n$ and diameter $d$. For integers satisfying $d\geq 3$ and $d\neq 4$, write $S_n(1^{n-d},(d-1)^{(1)})$ for the star-like tree specified by the source.

Engbers–Erey's conjecture. The average size of maximal matchings, $avm(T)$, is uniquely minimized by

$$S_n(1^{n-d},(d-1)^{(1)}).$$

This conjecture concerns the extremal value of the average size of maximal matchings among trees with prescribed order and diameter. The cited context states that the corresponding extremal graphs were known for trees of diameter at most five, while this assertion addresses the remaining specified diameters; its resolution is not given here.