The conjecture for alpha in the intermediate partite range

Let $\alpha(k,r)$ be the parameter defined as the minimum number of edges between a specified independent transversal and its complement over all $K_r$-partite-saturated $k$-partite graphs, with $k\ge r\ge 3$. The conjecture for $\alpha(k,r)$. For

$$5\le r\le k\le 2r-4,$$

one has

$$\alpha(k,r)=(k-1)(4r-k-6).$$

The authors present this as the expected correct value of the upper bound in this range; the supplied text does not establish it.