Uniqueness conjecture for the Slater state in the ranges of the symmetric projections

Let $d$ parties share the multiqudit Slater state $\ket{\Psi_{\mathrm S}}$, and let $S_\mu$ be the symmetric projections defined from the measurement construction. Uniqueness conjecture. The Slater state $\ket{\Psi_{\mathrm S}}$ is the only state in the ranges of $S_\mu$. This conjecture is presented as a stronger version of the preceding proposition on stabilizing the Slater state and is used to derive the full self-testing result; its resolution is not given here.