Minimal block-length conjecture for qubit permutation-invariant codes

Let a qubit permutation-invariant quantum error-correcting code correct errors of weight at most $t$, and let $n_{\rm \min}(t)$ denote the smallest block length among such codes. The code distance is $d=2t+1$. Minimal qubit PI-code scaling conjecture. Minimal qubit permutation-invariant codes have

$$n_{\rm \min}(t)=3t^2+3t+1.$$

Equivalently, all qubit permutation-invariant quantum error-correcting codes satisfy

$$d\leq \frac{\sqrt{12n-3}}{3},$$

with equality for minimal permutation-invariant codes. This numerical conjecture would rule out linear-distance qubit PI codes, including good qLDPC families, even when encoding one logical qubit.