Linear bound for detecting failure of optimal domination

Let $p$ be the prime defining the quaternion algebra $B_p$, and let $\mathcal{O},\mathcal{O}'\subset B_p$ be maximal orders of different types. For an integer $b>0$, say that $\mathcal{O}'^T$ optimally dominates $\mathcal{O}^T$ up to $b$ when

$$\theta'_{\mathcal{O}^T}(k)\leq\theta'_{\mathcal{O}'^T}(k)$$

for all integers $k\leq b$. Linear-bound conjecture. There exists a bound $b=O(p)$ such that, for all maximal orders $\mathcal{O},\mathcal{O}'\subset B_p$ of different types, $\mathcal{O}'^T$ does not optimally dominate $\mathcal{O}^T$ up to $b$. This would give a linear bound on the search needed to distinguish maximal orders of different types and establish termination with controlled running time.