The spiders $S(4m+2,\,2m,\,1)$ are $e$-positive

Using the composition method, we establish the $e$-positivity of spiders of the form $S(4m+2,\, 2m,\, 1)$, as conjectured by Aliniaeifard, van Willigenburg and Wang. Following a divide-and-conquer strategy, we group one or two positive-coefficient $e_J$-terms with each negative-coefficient $e_I$-ter