Neighborly vertex-transitive triangulations of balanced sphere products

Let $d$ be the dimension of a triangulation, and let $S^{\lceil d/2\rceil}\times S^{\lfloor d/2\rfloor}$ be the corresponding product of spheres. A centrally $(\lfloor d/2\rfloor+1)$-neighborly triangulation is a centrally symmetric triangulation containing the prescribed crosspolytope skeleton, and a vertex-transitive dihedral group action is a dihedral group action transitive on the vertices.

Neighborly sphere-product conjecture. There is a centrally $(\lfloor d/2\rfloor+1)$-neighborly combinatorial triangulation of every product $S^{\lceil d/2\rceil}\times S^{\lfloor d/2\rfloor}$ with a vertex-transitive dihedral group action on $n=2d+4$ vertices.

The source proves existence for a listed finite collection of sphere products, including the balanced products in dimensions represented by the table, but does not establish the assertion for every $d$. Thus the general statement remains open.