Quasi-embeddings of polyhedra and compacta in products of dendrites

Let $P$ be a compact $n$-polyhedron. A quasi-embedding of $P$ in a product of $n$ dendrites means the approximation notion used in the source, namely the existence of arbitrarily fine maps from $P$ into that product. Let $I$ be the closed interval. Quasi-embedding conjecture. (a) If $P$ quasi-embeds in a product of $n$ dendrites, then $P\times I$ embeds in a product of $n+1$ trees. (b) The same conclusion holds when $P$ is a co-locally contractible $n$-dimensional compactum. This proposes a stabilization principle converting quasi-embeddability by dendrites into genuine embeddability by trees; the source presents it as an open problem.