The sigma-weight conjecture for binary linear codes

Let $C$ be a binary linear code, and let $\sigma$ be a permutation in its permutation automorphism group $\operatorname{PAut}(C)$. Let $E_\sigma(C)$ denote the associated $\sigma$-fixed subcode, and call it a non-distinct $\sigma$-weight code when its codewords have the corresponding non-distinct $\sigma$-weights. Sigma-weight conjecture. If

$$\operatorname{PAut}(C)=\langle\sigma\rangle,$$

then $E_\sigma(C)$ is a non-distinct $\sigma$-weight code. The conjecture is motivated by the examples presented in the paper, but no proof or resolution is supplied.