Harada–Norton and Held group characterization by linear supersingular factors

Let $HN$ be the Harada–Norton group and $He$ the Held group, with orders

$$\#HN=2^{14}\cdot3^{6}\cdot5^{6}\cdot7\cdot11\cdot19,\qquad \#He=2^{10}\cdot3^{3}\cdot5^{2}\cdot7^{3}\cdot17.$$

Let $L^{(5*)}(p)$ and $L^{(7*)}(p)$ denote the numbers of linear factors of the corresponding supersingular polynomials.

Sporadic-group linear-factor conjecture. For every prime $p$,

$$\deg ss^{(5*)}_{p}(X)=L^{(5*)}(p)\iff p\mid\#HN,$$

and

$$\deg ss^{(7*)}_{p}(X)=L^{(7*)}(p)\iff p\mid\#He.$$

This conjecture proposes an analogue of the level-1 characterization of primes by the complete splitting of supersingular polynomials, with the Harada–Norton and Held groups governing levels 5 and 7. The source gives no resolution evidence.