Exponential resistance growth conjecture for square grid graphs

Let $G=P_n\square P_n$ be the $n\times n$ grid graph with $n^2$ vertices, and let $H=P_{n+1}\square P_{n+1}$ be the $(n+1)\times(n+1)$ grid graph with $(n+1)^2$ vertices. Let $r_G(a,b)$ and $r_H(a,b)$ denote the resistance distances between the indicated vertices, and let $r_n(a,b)$ denote the corresponding resistance distance in the $n\times n$ grid. Square grid resistance growth conjecture.

$$\lim_{n\rightarrow \infty}\left(\exp(r_H(a,b))-\exp(r_G(a,b))\right)=C>0.$$

Moreover,

$$\lim_{n\rightarrow \infty}r_n(a,b)=\infty.$$

The conjecture asserts exponential growth of resistance distance under successive square-grid enlargements, together with divergence of the resistance distance. The source gives empirical evidence but does not establish the claim.