Triangular-lattice q2 convergence conjecture

Let $q_2(L)$ denote the crossing of the two largest eigenvalues in the $\ell=2$ sector for a triangular-lattice strip of width $L$ with fully periodic boundary conditions, and let $q_{\rm B}\simeq3.8196717312$ be Baxter's equimodularity value. Triangular-lattice $q_2$ convergence conjecture. The limit

$$q_2(\mathrm{tri})=\lim_{L\to\infty}q_2(L)$$

exists and equals $q_{\rm B}$; moreover, $q_2(3p)$ is monotonically decreasing in $p$. Numerical fits support the claim and imply a common limiting value with $q_0(\mathrm{tri})$, but no proof is given.