Asymptotic resistance increment conjecture for straight linear K-trees

Let $G$ be the straight linear $K$-tree with $n$ vertices and parameter $k\geq 1$, and let $H$ be the straight linear $K$-tree with $n+1$ vertices. Write $r_G(u,v)$ for the resistance distance between vertices $u$ and $v$ in $G$. Straight linear $K$-tree resistance conjecture.

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

The conjecture predicts a positive limiting increment in the maximal effective resistance between the extremal vertices of successive straight linear $K$-trees. The source reports that no closed formula is known in general, although empirical evidence supports the claim.