The BCH weight-enumerator integrality conjecture

Let $m$ and $t$ be integers, and let $C$ be obtained by adding an overall parity check to the primitive BCH code of length $2^m-1$ and designed distance $2t-1$. Thus $C$ has length $n=2^m$ and minimal distance $d\ge 2t$. Let $d'$ be the smallest power of $2$ satisfying $d'\ge 2t$, and let $\mathcal{P}_s$ denote the class of generating functions whose $s$th roots have integral coefficients. The BCH weight-enumerator integrality conjecture. The weight enumerator of $C$ belongs to

$$\mathcal{P}_{2^m/d'}.$$

This is presented as a conjectural analogue of the theorem for Reed–Muller codes; the source gives no proof or resolution.