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

For positive integers $m,n$, let $S(a,b,c)$ denote the spider formed from three paths of lengths $a,b,c$ 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 generalized conjecture. For any integers $m,n\ge 1$, the spider $S(n(n!m+1),\,n!m,\,1)$ is $e$-positive. The source presents this as a further conjecture and gives no resolution.