Thompson group characterization for class 3C supersingular polynomials

Let $Th$ be the Thompson sporadic simple group, and let $ss_{p}^{(3C)}(X)$ be the class-3C supersingular polynomial with $L^{(3C)}(p)$ linear-factor count.

Thompson-group conjecture. For every prime $p$,

$$\deg ss_{p}^{(3C)}(X)=L^{(3C)}(p)\iff p\mid\#Th.$$

The conjecture characterizes the primes for which the class-3C supersingular polynomial splits completely into linear factors using the prime divisors of the Thompson group order. The source gives no resolution evidence.