Missing-value conjecture for the next-to-middle missing-face number of neighborly spheres

Let $k\geq 2$, and let $\Delta$ be a neighborly $2k$-sphere with $n$ vertices. Write $m_{k+1}(\Delta)$ for its number of missing faces of dimension $k+1$. Missing-value conjecture. One has

$$m_{k+1}(\Delta)\neq \binom{n-k-3}{k}-1.$$

Furthermore, for $k\geq 3$, $n$ sufficiently large, and every integer $m$ with

$$0\leq m\leq \binom{n-k-3}{k},\qquad m\neq \binom{n-k-3}{k}-1,$$

there exists a neighborly $2k$-sphere with $n$ vertices and $m_{k+1}=m$. The conjecture specifies the sole excluded value in the proposed range of possible values of $m_{k+1}$ for sufficiently large neighborly spheres; the supplied text gives no resolution.