Dahlberg–She–van Willigenburg's non-e-positivity conjecture for trees

Let $T$ be a tree, let $\Delta(T)$ denote its maximum degree, and say that $T$ is $e$-positive when its chromatic symmetric function $X_T$ has a nonnegative expansion in the elementary symmetric-function basis. Dahlberg–She–van Willigenburg's conjecture. If

$$\Delta(T)\geq 4,$$

then $T$ is not $e$-positive. This conjecture links the structure of a tree to positivity properties of its chromatic symmetric function. It was verified for all trees with maximum degree at least $5$ and for spiders with maximum degree $4$, but the general case remains open.