Maximum-degree conjecture for trees with a given neighbor-locating-chromatic number

Let $k\geq 2$ and let $T$ be a tree. Write $\chi _{_{NL}}(T)$ for the neighbor-locating-chromatic number of $T$, and let $\Delta(T)$ denote its maximum degree. Maximum-degree conjecture. If

$$\chi _{_{NL}}(T)=k,$$

then

$$\Delta(T)\leq (k-1)^2.$$

Moreover, this bound is tight for every integer $k\geq 2$. The preceding discussion shows that the available general bound is not tight in some cases and motivates this proposed sharp bound; its validity and tightness are left as an open problem.