Conjecture on regular, physical singular, and strange solutions for the spin-5 Heisenberg chain

Let $s$ be the spin, $N$ the system size, and $ell$ the number of Bethe roots. A solution is called regular if it is a regular solution of the Bethe ansatz equations, physical singular if it is a singular solution satisfying the physicality condition, and strange if some components coincide and hence violate the Pauli principle. Define

$$\lambda^{(\ell)}_0=(\underbrace{0,\ldots,0}_{\ell}),\qquad \text{if }N\equiv\ell-1\pmod 2,$$

and

$$\lambda^{(\ell)}_\pm=(\underbrace{\pm s,\ldots,\pm s}_{\ell}),\qquad \text{if }N\equiv 2\ell-2\pmod 4.$$

Let $\mathcal{N}_{\mathrm{strange}}(N,\ell)$ and $\mathcal{N}_{\mathrm{sp}}(N,\ell)$ denote, respectively, the total numbers of strange and physical singular solutions corresponding to $N$ and $\ell$. The conjecture. (1) If $2s\equiv1\pmod2$, the total number of states consists of regular or physical singular solutions, except possibly for sporadic physical solutions; such solutions do occur, namely $\lambda^{(2)}_0$ for $N=3$ and $\lambda^{(2)}_\pm$ for $N=6$. (2) If $2s\equiv0\pmod2$, then for odd $\ell$ the total number of states is a union of regular and physical singular solutions, while for even $\ell$ the total number of solutions is a union of regular and strange solutions. (3) If $2s\equiv0\pmod2$, then

$$\mathcal{N}_{\mathrm{strange}}(2N,2\ell)=\mathcal{N}_{\mathrm{sp}}(2N-1,2\ell-1).$$

The conjecture is based on numerical data for the spin-$s$ generalized Heisenberg chain. The exceptional sporadic physical solutions are explicitly known, while the proposed classification and counting relation are not established in the source.