Dahlberg–Shepherd conjecture on the maximum degree of e-positive trees

A tree is a connected acyclic graph. A tree is $e$-positive when the expansion of its chromatic symmetric function in the elementary symmetric-function basis has only nonnegative coefficients.

Dahlberg–Shepherd conjecture. Any tree with a vertex of degree at least $4$ is not $e$-positive.

The source notes that every $n$-vertex $e$-positive tree has degree at most $\log_2 n$, and that positivity has been proved impossible when a vertex has degree at least $6$. Thus the degree-$4$ and degree-$5$ cases remain relevant to the conjecture.