Resistance increment conjecture for straight linear k-trees

Let $G$ be the straight linear $k$-tree with $k\geq 1$ and $n$ vertices, and let $H$ be the straight linear $k$-tree with $n+1$ vertices. Let $r_G$ and $r_H$ denote resistance distance in these graphs. Resistance increment conjecture.

$$\lim_{n\rightarrow\infty}\bigl(r_H(1,n+1)-r_G(1,n)\bigr)=\frac{6}{k(k+1)(2k+1)}.$$

For $k=2$, the paper proves the corresponding increment is $1/5$ and hence the end-to-end resistance is unbounded. The formula for general $k$ is supported by empirical evidence and remains open.