The quasi-heredity criterion for odd-rank algebras of type $B$

Let $n$ and $r$ be positive integers, let $q$ be the parameter, let $Q$ be the second parameter, and let $\mathcal{L}^n(2r-1)$ denote the algebra under consideration. Quasi-heredity conjecture. The algebra $\mathcal{L}^n(2r-1)$ is quasi-hereditary if and only if $Q\neq -q^k$ for all $k$ satisfying

$$\frac{4-n}{2}\leq k<\min(n,r).$$

This is proposed as a natural extension of the preceding results to smaller values of $m$. The surrounding discussion notes that quasi-heredity can occur for $m<2n+1$ even when the earlier condition holds, while the criterion is presented as a guess and no resolution is supplied here.