Totally asymmetric SIC existence conjecture over finite fields

Let a SIC be a system of $d^2$ equiangular lines, represented here by vectors over a finite field extension as in the paper. A SIC is totally asymmetric when its automorphism group is trivial. Totally asymmetric SIC existence conjecture. Totally asymmetric SICs exist in infinitely many dimensions over various finite fields. In particular, for infinitely many choices of $d$, there is a modular Hadamard matrix $H$ of order $d$ with trivial $\operatorname{Aut}_s(\tilde{H})$, and with $d\equiv 8\bmod p$ for some prime $p\equiv3\bmod4$, such that the associated SIC in $\mathbb{F}_{p^2}^d$ has trivial weak automorphism group. The conjecture is motivated by the empirical abundance of asymmetric Hadamard matrices and examples of totally asymmetric SICs; the abstract also formulates the weaker claim that such SICs exist in infinitely many dimensions, while the stated construction-based version remains unproved.