Even-width square-lattice strip conjecture for the thermodynamic crossing

Let $q_0(L)$ denote the maximum real value in the limiting curve for a square-lattice strip of width $L$ with fully periodic boundary conditions, and let $q_{\rm c}(\mathrm{sq})=3$. Even-width square-lattice conjecture. For widths $L=2p$, the limit

$$q_0(\mathrm{sq})=\lim_{p\to\infty}q_0(L)$$

exists and equals

$$q_0(\mathrm{sq})=q_{\rm c}(\mathrm{sq})=3.$$

Numerical fits give $q_0(\mathrm{sq})=2.999\pm0.012$ and support the claim, but the source does not report a proof.