The Sylow q-subquasigroup generator conjecture

Let $g$ generate the maximal multiplicative subgroupoid $\mathbb{E}^*_{(p-1)^2}$ of an entropoid, with $p\geq 11$. Define

$$g_q=g*(g*(g*((g*g)*g))).$$

The Sylow generator conjecture. The element $g_q$ generates the Sylow $q$-subquasigroup $\mathbb{E}^*_{q^2}$. This would provide an explicit way to obtain a generator of the Sylow subquasigroup from a generator of the maximal one; the paper leaves the assertion unproved.