Facet- and vertex-minimality conjecture for the degree construction

Let $n$ be the dimension and let $k$ be the integer used in Theorem. Consider the triangulated $n$-sphere constructed there, together with its simplicial degree $k$ map to the standard triangulated sphere.

Minimality conjecture. The triangulated $n$-sphere constructed in Theorem is facet-minimal, and it is vertex-minimal as well.

The question concerns whether the construction gives optimal triangulations simultaneously with respect to the numbers of facets and vertices. The source presents this as a conjecture suggested by the facet-preimage behavior in the proof of the theorem; no resolution is given here.