Completely independent spanning trees in the hypercube

We say two spanning trees of a graph are completely independent if their edge sets are disjoint, and for each pair of vertices, the paths between them in each spanning tree do not have any other vertex in common. Pai and Chang constructed two such spanning trees in the hypercube $Q_n$ for sufficiently large $n$, while Kandekar and Mane recently showed there are $3$ pairwise completely independent spanning trees in hypercubes $Q_n$ for sufficiently large $n$. We prove that for each $k$, there exist $k$ completely independent spanning trees in $Q_n$ for sufficiently large $n$. In fact, we show that there are $(\frac{1}{12}+o(1))n$ spanning trees in $Q_n$, each with diameter $(2+o(1))n$. As the minimal diameter of any spanning tree of $Q_n$ is $2n-1$, this diameter is asymptotically optimal. We prove a similar result for the powers $H^n$ of any fixed graph $H$.