  We study the Bethe Ansatz formula for the superconformal index, in the case
of 4d $\mathcal{N}=4$ super-Yang-Mills with gauge group $SU(N)$. We observe
that not all solutions to the Bethe Ansatz Equations (BAEs) contribute to the
index, and thus formulate "reduced BAEs" such that all and only their solutions
contribute. We then propose, sharpening a conjecture of Arabi Ardehali et al.
[arXiv:1912.04169], that there is a one-to-one correspondence between branches
of solutions to the reduced BAEs and vacua of the 4d $\mathcal{N}=1^*$ theory.
We test the proposal in the case of $SU(2)$ and $SU(3)$. In the case of
$SU(3)$, we confirm that there is a continuous family of solutions, whose
contribution to the index is non-vanishing.
