Sharpness of the edge-distinguishing bound for infinite locally finite star-free graphs

Let $n$ be a natural number greater than three. A connected infinite locally finite graph is a graph with infinitely many vertices, finite vertex degrees, and a path between every pair of vertices; for a graph $G$, let $D'(G)$ denote its distinguishing index, the least number of colours in an edge colouring preserved only by the identity automorphism.

Sharpness conjecture. For every natural number $n$ greater than three there exists a connected infinite locally finite graph $G$ such that

$$D'(G)=n-1.$$

The conjecture asks whether the upper bound $D'(G)\leqslant n-1$ for connected locally finite $K_{1,n}$-free graphs is attained for every $n\geqslant4$ in the infinite case. The paper notes that the bound is attained for finite graphs and for infinite locally finite $K_{1,3}$-free graphs, while the cases $n>3$ remain open.