The DEDHP hardness conjecture in Sylow q-subquasigroups

Let $\mathbb{E}_{p^2}$ be an entropoid with $p=2q+1$ a safe prime, and let $g_q$ generate its Sylow $q$-subquasigroup $\mathbb{E}^*_{q^2}$. The Sylow DEDHP hardness conjecture. No algorithm $\mathcal{A}$ solves DEDHP in $\mathbb{E}^*_{q^2}$ with significantly greater advantage than uniformly random guessing. The claim is presented as a plausible cryptographic conjecture after showing that DEDHP is efficiently solvable in the maximal quasigroup; no resolution is given.