Minor monotonicity of embeddability in spheres

Let $H$ and $K$ be simplicial complexes, and write $H<K$ when $H$ is a minor of $K$, obtained by admissible contractions and deletions. Let $m$ be a nonnegative integer. Minor monotonicity of embeddability. If $H<K$ and $K$ is embeddable in the $m$-sphere, then $H$ is embeddable in the $m$-sphere. This would make embeddability in a fixed-dimensional sphere minor-monotone; the source notes that it would imply the preceding obstruction-based corollary, but gives no resolution of the conjecture.