Asymptotic resistance increment conjecture for block tower graphs

Let $G$ be the block tower graph $C_4\square P_n$, the Cartesian product of the $4$-cycle and the path of length $n$, with $4n$ vertices, and let $H$ be the block tower graph $C_4\square P_{n+1}$ with $4n+4$ vertices. Write $r_G(u,v)$ for the resistance distance between vertices $u$ and $v$ in $G$. Block tower resistance conjecture.

$$\lim_{n\rightarrow \infty} \bigl(r_H(1,4n+3)-r_G(1,4n-1)\bigr)=\frac{1}{4}.$$

The conjecture predicts a positive limiting increment in the effective resistance between corresponding extremal vertices of successive block tower graphs. The source presents it as an empirically motivated conjecture and gives no closed formula in general.