Conjectural factor counts for the Hasse invariant of the Tate normal form $E_7$

Let $l\neq 2,3,7$ be prime. Let $\hat H_{7,l}(x)$ be the Hasse-invariant polynomial of the Tate normal form $E_7$ over $\mathbb F_l$, and let $h(-l)$ be the relevant class number. Conjectural factor-count formulas. If $l\equiv6\pmod 7$, the number $N_1$ of distinct linear factors dividing $\hat H_{7,l}(x)$ is

$$N_1=\begin{cases}3h(-l),&l\equiv1\pmod4,\\3\left(3-\left(\frac2l\right)\right)h(-l),&l\equiv3\pmod4.\end{cases}$$

If $l\equiv3,5\pmod7$, the number $N_3$ of distinct irreducible cubic factors is

$$N_3=\begin{cases}h(-l),&l\equiv1\pmod4,\left(3-\left(\frac2l\right)\right)h(-l),&l\equiv3\pmod4.\end{cases}$$

The first count is six times, and the second twice, the number of supersingular $j$-invariants lying in $\mathbb F_l$. These formulas were stated in the cited source as conjectures, although the surrounding paper says the first is proved later.