The average-degree rainbow path conjecture for subgraphs of hypercubes

Let $Q_n$ be the $n$-dimensional hypercube, and let $G$ be a subgraph of $Q_n$. A proper edge-coloring of $G$ is one in which incident edges receive distinct colors, and a rainbow copy of a graph is a copy whose edges receive pairwise distinct colors.

Average-degree rainbow path conjecture. If $G$ has average degree at least $d$, then every proper edge-coloring of $G$ contains a rainbow copy of the path on $d$ edges.

This is an average-degree analogue of the Erdős–Sós conjecture and a rainbow conjecture proposed in the cited work. Even for subgraphs of hypercubes, the assertion remains open for $d\ge 5$.