Conjecture on odd-characteristic BCH codes with prescribed dual distance and locality

Let $q=p^s$, where $p$ is an odd prime and $s\ge 2$, and set $h=\frac{p-1}{2}$. Let $\mathcal{C}_{(q,q+1,3,h)}$ denote the BCH code in question. BCH-code conjecture. The code $\mathcal{C}_{(q,q+1,3,h)}$ is an AMDS code with parameters $[q+1,q-3,4]$, while its dual has parameters $[q+1,4,q-p]$. Moreover, $\mathcal{C}_{(q,q+1,3,h)}$ is a $d$-optimal and $k$-optimal LRC with locality $q-p-1$, and its dual is an LRC with locality $3$. The claim is motivated by computations for even $s$ and several values of $p$; the preceding theorem establishes related parameter and locality results, but the asserted exact dual distance and locality remain unproved in the stated generality.