e-positivity conjecture for the sporadic spiders S(b²+b,b,1)

For integers $a,b$ with $a\ge b\ge 2$, let $S(a,b,1)$ be the spider formed from paths of lengths corresponding to the three parameters, with a common endpoint. A graph is $e$-positive when its chromatic symmetric function has a nonnegative expansion in the elementary symmetric-function basis.

Spider e-positivity conjecture. If $b$ is even and $b\not\equiv2\pmod3$, then the spider $S(b^2+b,\,b,\,1)$ is $e$-positive.

The conjecture is proposed because $a=b^2+b$ is a sporadic case in the paper’s bounds for $S(a,b,1)$. The source reports that it has been checked for $b\in\{4,6\}$, while no general proof is stated.