Blažek–Koman conjecture for the $k$-page book crossing number of $K_n$

Let $K_n$ be the complete graph on $n$ vertices, let $k$ be a positive integer, and let $\nu_k(K_n)$ denote its $k$-page book crossing number. Define

$$F(r,n):=\frac{r}{24}(r^2-3r+2)(2n-3-r)$$

and

$$Z_k(n):=(n\bmod k)\cdot F\left(\left\lfloor\frac{n}{k}\right\rfloor+1,n\right)+(k-(n\bmod k))\cdot F\left(\left\lfloor\frac{n}{k}\right\rfloor,n\right).$$

Blažek–Koman conjecture. For any positive integers $k$ and $n$,

$$\nu_k(K_n)=Z_k(n).$$

The quantity $Z_k(n)$ is the crossing count of standard constructions generalizing the Blažek–Koman construction. These constructions are widely believed to be asymptotically correct, but the exact equality is presented here as a conjecture.