Rainbow minimum-degree Erdős–Sós conjecture for hypercubes

For a graph $G$ and a graph $F$, let $\delta^*(G,F)$ be the maximum minimum degree of a subgraph $H\subset G$ that has a proper edge-coloring with no rainbow copy of $F$. Let $Q_n$ denote the $n$-dimensional hypercube, and let $T$ be a tree with $k$ edges. Rainbow minimum-degree Erdős–Sós conjecture for hypercubes.

$$\delta^*(Q_n,T)=k-1$$

for all trees $T$ on $k$ edges. This is proposed as an alternative to complete graphs, where the corresponding rainbow minimum-degree quantity can depend on the structure of the tree. The supplied text gives no resolution.