Maximum-degree-four tree e-positivity conjecture

Prove that every tree with maximum degree at least 4 is not e-positive.

Background

Dahlberg, She, and van Willigenburg proposed this conjecture after proving non-e-positivity for trees whose maximum degree is sufficiently large relative to their number of vertices. The paper records subsequent progress: trees of maximum degree at least 6, and later at least 5, were shown to be non-e-positive, but the degree-four case remained unresolved in the cited context. The results of the paper provide additional evidence toward this conjecture by ruling out e-positivity for further classes of trees with maximum degree 3 or 4.