Exact rainbow path problem for regular digraphs

Determine whether every d-regular digraph that is properly edge-coloured with d colours contains a directed rainbow path with d−1 edges.

Background

A positive solution would imply Graham’s conjecture, the Alspach–Liversidge conjecture, and their nonabelian analogues through coloured Cayley digraphs. The paper proves an asymptotic result only in the dense regime, leaving the exact general question unresolved.

References

Let $G$ be a $d$-regular digraph properly edge-coloured with $d$ colours. Does $G$ contain a directed rainbow path with $d-1$ edges?

— Towards Graham's rearrangement conjecture via rainbow paths  (2503.01825 - Bucić et al., 3 Mar 2025) in Problem 3, Section 1; discussed again in Section 6, “Rainbow walks in regular digraphs”