Chevyrev–Galbraith's optimal domination conjecture for maximal quaternion orders

Let $p$ be a prime and let $B_p$ be the quaternion algebra over $\mathbb{Q}$ ramified precisely at $p$ and $\infty$. Let $\mathcal{O}$ and $\mathcal{O}'$ be maximal orders of $B_p$. For a maximal order $\mathcal{O}$, write $\mathcal{O}^{T}$ for its associated trace-zero lattice, and let $a_{\mathcal{O}^{T}}(n)$ denote the relevant Fourier coefficient. Say that $\mathcal{O}'^{T}$ optimally dominates $\mathcal{O}^{T}$ if

$$a_{\mathcal{O}'^{T}}(n)\geq a_{\mathcal{O}^{T}}(n)$$

for every $n\in\mathbb{N}_0$. Chevyrev–Galbraith's conjecture. If $\mathcal{O}'^{T}$ optimally dominates $\mathcal{O}^{T}$, then $\mathcal{O}$ and $\mathcal{O}'$ are of the same type. This means that they are conjugate by a nonzero element of $B_p$. The conjecture is precisely the condition expected to ensure termination of the Chevyrev–Galbraith algorithm for constructing a supersingular elliptic curve with a prescribed maximal endomorphism ring; its resolution is not established in the supplied text.