Eulerian edge-refinement conjecture for sphere embeddings

Let $G$ be a $d$-sphere with $d\geq 2$ and let $H$ be a $(d+1)$-sphere containing $G$ as a subgraph. An edge refinement inserts a vertex into an edge and joins it to the common neighbors of the edge's endpoints. A graph is Eulerian when the relevant simplex degrees are even.

Eulerian edge-refinement conjecture. It is possible to edge refine $H$, using only edges different from those of $G$, so that the modified $H$ becomes Eulerian.

The conjecture is proposed as a sufficient condition for the sphere coloring conjecture: an Eulerian host would provide a coloring of the embedded sphere with $d+1$ or $d+2$ colors. The source gives no resolution and notes that the higher-dimensional cases remain to be investigated.