Conjecture that Szöllősi-family matrices cannot belong to MUB-quadruplets

Let $X(\alpha)$ denote the Szöllősi family of complex Hadamard matrices referred to in the source. A matrix $S\in X(\alpha)$ is considered part of an MUB-quadruplet when it occurs among the transition matrices associated with four mutually unbiased bases.

Szöllősi-family non-extendability conjecture. Any matrix $S\in X(\alpha)$ in the Szöllősi family cannot be part of an MUB-quadruplet.

The source explains that this claim was previously treated as a proven fact, but a proof in the cited literature was found to be incorrect. It is therefore formulated here as a conjecture and is intended to support the argument that the maximal number of mutually unbiased bases in dimension six is three.