Fine-grained complexity of induced non-shortest-path detection

Determine the fine-grained complexity of deciding whether an induced non-shortest path (a uv-trail) exists between two designated vertices at distance at least three, including whether the problem admits an algorithm substantially faster than the currently known matrix-multiplication-based bounds.

Background

The paper introduces induced non-shortest paths, or uv-trails, as induced paths between two vertices that are not shortest paths. Prior work gave algorithms for a single pair, most recently in time O(n^{2\omega}), while the paper proves a conditional lower-bound connection to triangle detection for pairs at distance at least three. Despite this progress, the authors leave the broader fine-grained complexity of the problem unresolved.