  We provide a conjecture for the following two quantities related with the
spin-$\frac{1}{2}$ isotropic Heisenberg model defined over rings of even
lengths: (i) the number of the solutions to the Bethe ansatz equations which
correspond to non-zero Bethe vectors; (ii) the number of physical singular
solutions of the Bethe ansatz equations in the sense of Nepomechie-Wang. The
conjecture is based on a natural relationship between the solutions to the
Bethe ansatz equations and the rigged configurations.
