The automorphism-group conjecture for extremal self-dual binary codes of length 72

Let $C$ be a binary linear self-dual $[72,36,16]$ code, and let $\sigma$ be a permutation in its permutation automorphism group $\operatorname{PAut}(C)$. Automorphism-group conjecture. If

$$\sigma\in\operatorname{PAut}(C),$$

then

$$\operatorname{PAut}(C)\neq\langle\sigma\rangle.$$

Equivalently, the permutation automorphism group cannot be generated by $\sigma$; the source supplies no resolution.