Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series $T_{31A}$

We conjecture a positive four-shift identity for rank-two Nahm sums with denominator steps $(1,31)$, giving a mixed-base realization of the Monster McKay--Thompson series $T_{31A}$ through the classical Rogers--Ramanujan function $U(1,31)$. A theta dissection directly derives the modular target. For