Byott's solvability conjecture for finite groups with Hopf-Galois structures

Let $\Gamma$ and $\Delta$ be finite groups of the same order. Write $e(\Gamma,\Delta)$ for the number of Hopf-Galois structures on a Galois extension with Galois group $\Gamma$ and underlying group $\Delta$. Byott's solvability conjecture. If $\Gamma$ is insolvable and $e(\Gamma,\Delta)\neq0$, then $\Delta$ is also insolvable. This conjecture asserts that the existence of a Hopf-Galois structure cannot transfer insolvability from the Galois group to a solvable underlying group. The surrounding discussion presents it as a remark due to N. P. Byott; no resolution is given here.