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

By using the composition method, we establish the $e$-positivity of spiders of the form $S(4m+2,\, 2m,\, 1)$, which was conjectured by Aliniaeifard, van Willigenburg and Wang. Following the divide-and-conquer strategy, we group one or two $e_J$-terms that have positive coefficients with each $e_I$-term that has a negative coefficient, where the compositions $J$ are selected to be obtained by rearranging the parts of $I$, and show the positivity of the sum of those coefficients. Our main contribution is an explicit construction of the injection.