The dimension and vertex-number conjecture for complementary weak pseudomanifolds

A complementary weak pseudomanifold is a weak pseudomanifold whose simplicial complex satisfies complementarity. Let $M$ be a $d$-dimensional complementary weak pseudomanifold with $n$ vertices. Dimension and vertex-number conjecture. If such an $M$ exists, then $d$ is even and

$$n=3d/2+3.$$

Known results establish this conclusion under the additional assumptions $n\leq d+6$ or $d\leq 6$, and show that in dimensions at most $5$ the pseudomanifold is a combinatorial manifold. The conjecture asserts the corresponding restrictions in all dimensions.