Exponential resistance growth conjecture for triangular grid graphs

Let $T_n$ be the triangular grid graph shown in Figure~, with $n$ rows and $m=n^2$ cells. Let $a$ and $b$ be distinct vertices of degree $2$, and let $r_n(a,b)$ be the resistance distance between $a$ and $b$ in $T_n$. Triangular grid resistance growth conjecture.

$$\lim_{n\rightarrow \infty}\left(\exp(r_{n+1}(a,b))-\exp(r_n(a,b))\right)=C>0.$$

Moreover,

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

The conjecture predicts exponential growth of resistance distance between fixed degree-$2$ vertices as the triangular grid expands, and in particular divergence of that resistance distance. The source offers empirical evidence but no proof.