Motivic symplectic cohomology distinguishes symplectic forms in a connected deformation class

Let $X_1=(X,\omega_1)$ and $X_2=(X,\omega_2)$ be symplectic manifolds on the same underlying manifold, where $\omega_1$ and $\omega_2$ are connected by a path of symplectic forms. For symplectic manifolds $Y$ and $M$, consider the groups $SH_{\bullet}^M(Y,X_1)$ and $SH_{\bullet}^M(Y,X_2)$. Distinguishing conjecture. One can choose some $Y$ and $M$ so that these groups distinguish $X_1$ and $X_2$. This would make the groups $SH_{\bullet}^M(Y,X)$ sensitive to the symplectic form within a connected deformation class. The source does not provide a resolution status for this claim, so it is recorded as open.