Conjecture that combining spider legs preserves $e$-positivity

Let $S=S(\a0\lambda_1,\ldots,\lambda_d)$ be a spider with $d$ legs, where the $\a0\lambda_i$ are the leg lengths. Let $S'$ be the spider with $d-1$ legs obtained by combining two legs into one leg whose length is their sum. A spider is $e$-positive when its chromatic symmetric function is a nonnegative linear combination of elementary symmetric functions.

Leg-combination conjecture. If $S$ is $e$-positive, then $S'$ is $e$-positive.

The conjecture is motivated by the fact that the paper's known $e$-positivity tests are preserved when two legs are combined. The Dahlberg–She–van Willigenburg conjecture would imply it if the only $e$-positive spiders had at most three legs, but the source leaves the general statement open.