Aliniaeifard, van Willigenburg and Wang's spider e-positivity conjecture

For a partition $ambda=ambda_1\dotsmambda_d$ of $n-1$ with $d\ge 3$, the spider $S(\lambda)$ is the $n$-vertex tree consisting of paths of lengths $\lambda_1,\dots,\lambda_d$ with a common end. A graph is $e$-positive when its chromatic symmetric function has nonnegative coefficients in the elementary symmetric-function basis. Aliniaeifard, van Willigenburg and Wang's conjecture. For any integer $m\ge 1$, the spider $S(4m+2,\,2m,\,1)$ is $e$-positive. The source attributes this conjecture to Aliniaeifard, van Willigenburg and Wang and gives no resolution.