Generalised Thue chromatic index bound for Jaco graphs

Let $J^{\ast}_n(f(x))$ be a Jaco graph, where

$$f(x)=mx+c,$$

with $x\in\mathbb{N}$, $m\geq 2$, and $c\geq 1$. Write $\pi(J^{\ast}_n(f(x)))$ for its Thue chromatic index, and let $\Delta(J^{\ast}_n(f(x)))$ and $\delta(J^{\ast}_n(f(x)))$ denote its maximum and minimum degrees, respectively.

Generalised Jaco graph conjecture. For every such Jaco graph,

$$\pi(J^{\ast}_n(f(x)))\leq \Delta(J^{\ast}_n(f(x)))+\delta(J^{\ast}_n(f(x))).$$

This conjecture generalises the preceding bound for the considered Jaco graphs by relating the Thue chromatic index to both the maximum and minimum degrees. The source provides no resolution, so the conjecture remains open.