Sparla's existence conjecture for neighborly triangulations of sphere products

Let $r\geq 1$. A centrally $(r+1)$-neighborly triangulation is a centrally symmetric triangulation containing the $r$-skeleton of the relevant crosspolytope boundary. Consider the sphere product $S^r\times S^r$.

Sparla's existence conjecture. There are centrally $(r+1)$-neighborly triangulations of $S^r\times S^r$ on $4r+4$ vertices.

The source motivates this claim from the equality case of Sparla's Euler characteristic bound: for $S^r\times S^r$ with $4r+4$ vertices, equality would force the $r$-skeleton of $\partial C_{2r+2}^{\Delta}$. The source does not state a general resolution of the existence claim.