The nonsquare supersingular lambda-invariant conjecture

Assume $N$ is a prime with $N\equiv 1\pmod 4$, let $L$ be the set of supersingular lambda invariants in $\mathbf{F}_{N^2}$, and let $H'$ denote the derivative of the polynomial $H$ introduced in the supersingular lambda-invariant construction. Nonsquare supersingular lambda-invariant conjecture. For every $\lambda\in L$, $H'(\lambda)$ is not a square in $\mathbf{F}_{N^2}^{\times}$. The assertion was checked numerically for $N<2000$; it fails when $N\equiv3\pmod4$, so the stated case remains conjectural.