Paley I automorphism conjecture for finite-field SICs

Let $q>11$ be a prime power with $q\equiv3\bmod4$, let $d=q+1$, and let $H$ be the Paley I Hadamard matrix of order $d$. Write $\tilde H$ for the associated object and $\operatorname{Aut}_s$ and $\operatorname{Aut}_w$ for the strong and weak automorphism groups used in the paper. Paley I automorphism conjecture. The group $\operatorname{Aut}_s(\tilde H)\cong P\Gamma L(2,q)$ has exactly two orbits on $[d]\times[d]$, namely $\{(i,i):i\in[d]\}$ and its complement; moreover, for every prime $p\equiv3\bmod4$ dividing $d-8$, the associated SIC $\{x_{ij}\}$ in $\mathbb F_{p^2}^d$ satisfies $\operatorname{Aut}_s(\{x_{ij}\})=\iota(\operatorname{Aut}_w(H))$ and $\operatorname{Aut}_w(\{x_{ij}\})=\operatorname{Aut}_s(\tilde H)$. The paper proves that $\operatorname{Aut}_s(\tilde H)$ contains a copy of $P\Gamma L(2,q)$ with the stated orbits, but does not prove the reverse inclusion; consequently the conjecture remains open.