The maximal-order containment bound for Bass orders

Let $E$ be a supersingular elliptic curve and let $\Lambda\subseteq\operatorname{End}(E)$ be a Bass order whose size is polynomial in $\log p$. Fix an integer $k\geq 0$ and assume that

$$\operatorname{discrd}(\Lambda)=O(p^k).$$

Maximal-order containment conjecture. For every $\epsilon>0$, the number of maximal orders containing $\Lambda$ is

$$O(p^{\epsilon}).$$

The paper proves this bound when $\operatorname{discrd}(\Lambda)$ is square-free and conjectures that it continues to hold without the square-free assumption, supporting a subpolynomial running time for the global step of the endomorphism-ring algorithm.