Puncture-code distance conjecture for cyclic quantum MDS codes

Let $q$ be a prime power and let $C^*=[q^2+1,q^2+1-d,d]_{q^2}$ be an $F_{q^2}$-linear (consta)cyclic MDS code. Let $P(C)$ denote its corresponding puncture code. Puncture-code distance conjecture. The puncture code $P(C)$ has parameters $[q^2+1,q^2+1-(d-1)^2,d']_q$, where

$$d'=\begin{cases}2(d-1)&\text{for }1<d\le q/2+1,\\(q+1)(d-1-\lfloor q/2\rfloor)&\text{for }q/2+1<d\le q,\ q\text{ odd},\q(d-\lfloor q/2\rfloor)&\text{for }q/2+1<d\le q,\ q\text{ even},\q^2+1&\text{for }d=q+1. \end{cases}$$

This is suggested by computational results for the corresponding puncture codes; no general proof is supplied.