Uniqueness conjecture for the SY class of a curve

Let $X$ be a smooth projective curve in the setting above, and let the SY torsor of $X$ parametrize the choices used to construct its SY bundles. Each element of this torsor determines an SY class in $\operatorname{Br}(X^{(2)})[2]$, namely the class of the corresponding SY bundle, which is trivial if and only if that bundle admits a section. Uniqueness conjecture for the SY class. There is only one SY class associated to $X$; equivalently, the construction is independent of the choice of an element of the SY torsor. The claim is presented as plausible but unverified; it is known when $k$ is finite or algebraically closed, with additional supporting evidence for supersingular elliptic curves.