Hypotheses regarding Baxter's $T-Q$ relation for the periodic XYZ chain

Baxter’s $T-Q$ relation for the periodic spin-$\frac{1}{2}$ XYZ chain is studied. We extensively perform numerical calculations for the $T-Q$ relation and the Bethe ansatz equations. Numerical based hypotheses are then proposed to answer some open questions regarding Baxter’s $T-Q$ relation and the XYZ chain.