Rainbow Erdős–Sós conjecture for hypercubes

For a graph $G$ and a graph $F$, let $\mathrm{ex^*}(G,F)$ be the maximum number of edges in a subgraph of $G$ that has a proper edge-coloring with no rainbow copy of $F$. The $n$-dimensional hypercube $Q_n$ has vertex set $\{0,1\}^n$, with two vertices adjacent when they differ in exactly one coordinate. Rainbow Erdős–Sós conjecture for hypercubes. For each $n\in\mathbb{N}$,

$$\mathrm{ex^*}(Q_n,T)=\mathrm{ex^*}(Q_n,K_{1,k})$$

for all trees $T$ on $k$ edges; equivalently, $Q_n$ satisfies the host-graph version of the rainbow Erdős–Sós question. This conjecture proposes that the rainbow extremal number on the hypercube depends only on the number of edges in the forbidden tree. The supplied text gives no resolution.