Flip-map conjectures for Bethe solutions and rigged configurations

Let $\mathrm{BA}(\ell)$ be the set of regular or physical singular Bethe solutions, and let $\iota:\mathrm{BA}(\ell)\to\mathrm{BA}(\ell)$ be defined by

$$(\lambda_1,\ldots,\lambda_\ell)\longmapsto(-\lambda_1,\ldots,-\lambda_\ell).$$

For the generalized Heisenberg chain on the length-$N$ tensor product of the spin-$s$ representation, set $\mu=(2s,\ldots,2s)$ with $N$ parts. Flip-map conjectures. (A) The map $\iota$ induces the flip map on rigged configurations. (B) If $N$ is even, then: (a) if $2s$ is odd and $\ell$ is even, physical singular solutions of $\mathrm{BAE}(N,\ell)$ correspond one-to-one with flip-invariant rigged configurations whose partition $\nu$ contains an odd number of even parts at least $2s+1$; (b) if $2s$ is even and $\ell$ is odd, they correspond one-to-one with flip-invariant rigged configurations whose partition $\nu$ contains an odd number of odd parts at least $2s+1$. (C) If $s=1/2$, $N$ is even, and $\ell$ is odd, physical singular solutions of $\mathrm{BAE}(N,\ell)$ correspond one-to-one with flip-invariant rigged configurations whose partition $\nu$ contains an odd number of even parts longer than $2s+1$, excluding the rigging

$$J_{k,1}=J_{k,2}=\cdots=J_{k,m_k}=\frac{P_k(\nu)}{2}$$

when $m_k(\nu)\geq3$ is odd and $P_k(\nu)>0$ is divisible by $4$. These conjectures refine the proposed bijection by describing its compatibility with the flip symmetry and predicting which configurations correspond to physical singular solutions in specified parity cases. The source gives no resolution status.