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The shrinking operation converts a hypergraph into a graph by choosing, from each hyperedge, two endvertices of a corresponding graph edge. A hypertree is a hypergraph which can be shrunk to a tree on the same vertex set. Klimosov\'a and Thomass\'e J. Combin. Theory Ser. B 156 (2022), 250--293 proved (as a tool to obtain their main result on edge-decompositions of graphs into paths of equal length) that any rank 3 hypertree T can be shrunk to a tree where the degree of each vertex is at least 1/100 times its degree in T. We prove a stronger and a more general bound, replacing the constant 1/100 with 1/2k when the rank is k. In place of entropy compression (used by Klimosov\'a and Thomass\'e), we use a hypergraph orientation lemma combined with a characterisation of edge-coloured graphs admitting rainbow spanning trees.
Hylasová et al. (Fri,) studied this question.