Aliniaeifard–van Willigenburg–Wang conjecture for a generalized family of spiders

For positive integers $n$ and $m$, let $S(a,b,c)$ denote the spider with three legs of lengths $a$, $b$, and $c$. A spider is $e$-positive when its chromatic symmetric function is a nonnegative linear combination of elementary symmetric functions.

Aliniaeifard–van Willigenburg–Wang conjecture. The family of spiders

$$S(n(n!m+1),n!m,1)$$

is $e$-positive.

This is the second related conjecture communicated to the authors by Aliniaeifard, van Willigenburg, and Wang. The source gives no resolution.