Refined Singleton-bound conjecture for quotient-projective-line quantum codes

Let $X=\mathbb{P}^1/G$ be a quotient of the projective line by a finite group $G$ with positive Euler characteristic $\chi(X)$. For a quantum stabilizer code constructed by the CSS method from self-orthogonal algebraic geometry codes on $X$, let $n$ be the number of rational points, $k$ the number of encoded qubits, and $d$ the minimum distance. For each point $p$, let $G_p$ be its stabilizer subgroup, and define

$$\epsilon=\frac{1}{2}\sum_p\left(1-\frac{1}{|G_p|}\right)>0.$$

Refined Singleton-bound conjecture. The minimum distance satisfies

$$d\leq\frac{n-k+2}{2}-\frac{\epsilon}{2}.$$

This is presented as a conjectural refinement of the quantum Singleton bound for quotient spaces, with partial-results discussion focused on quotients of the projective line and their twisted sectors; the source gives no proof or resolution of the general assertion.