Eulerian edge-refinement conjecture for discrete spheres

Let a $d$-sphere be a finite simple graph whose unit spheres are $(d-1)$-spheres and which becomes contractible after deleting a suitable vertex. An edge refinement is the operation of inserting a vertex into an edge and connecting it to the common neighbors of the edge's endpoints. A graph is Eulerian when the degrees of its relevant simplices are even.

Eulerian edge-refinement conjecture. $d$-spheres have edge refinements which are Eulerian.

The statement is proposed as a higher-dimensional extension of the result that every $2$-sphere admits edge refinements making it Eulerian. The source specifically says that the corresponding higher-dimensional bootstrapping remains to be established.