Conjectured convergence diagram for half-plane self-avoiding-walk measures

Let $\varphi^{m,\lambda,k}(y_0,y_1,\ldots,y_k)$ denote the corresponding function for the finite approximation indexed by $m$, let $T^m$ be the finite approximation of the half-plane self-avoiding-walk tree, and let $\varphi^{\mathbb H,\lambda,k}$ and $\varphi^{\lambda_c,k}$ denote the half-plane function and its critical-limit function. The parameters satisfy $\lambda>\lambda_c(T_{\mathbb H})$ along the upper horizontal limit and $\lambda\to\lambda_c(T^m)$ along the left vertical limit.

Convergence-diagram conjecture. The following convergence diagram holds:

\xymatrix{ \varphi^{m,\lambda,k}(y_0, y_1,\ldots,y_k)\ar[r]^{m \rightarrow +\infty}_{\lambda >\lambda_c(T_{\mathbb{H}})} \ar[d]_{\lambda \rightarrow \lambda_c(T^m)}& \varphi^{\mathbb{H},\lambda,k}(y_0, y_1,\ldots,y_k) \ar[d]^{\lambda\rightarrow \lambda_c}\\ \varphi^{m,\lambda_m,k}(y_0, y_1,\ldots,y_k) \ar[r]_{m\rightarrow +\infty}& \varphi^{\lambda_c,k}(y_0, y_1,\ldots,y_k) }

This asserts compatibility between taking the finite-tree limit and taking the critical-bias limit. The supplied text does not provide a proof or a resolution.