Mohar's conjecture on the crossing number of antipodal multipartite graphs

Let $M_{n,t}$ be the graph considered in the construction above, and write $k=\lfloor n/2\rfloor$. If a set $P'\subseteq P$ has strength $0$ and $t=|P\setminus P'|$, then the construction gives a drawing of $M_{n,t}$ with $H(n)-\tfrac{1}{2}t(k-1)(k-2)$ crossings. Mohar's conjecture. The crossing number of $M_{n,t}$ is equal to

$$H(n)-\tfrac{1}{2}t(k-1)(k-2).$$

This conjecture asserts that the displayed construction is optimal for these graphs; the supplied text gives no resolution, so its status is open.