The complete multipartite fractional-defect conjecture

For positive integers $m$, $a$, and $k$, let $K^{(m)}_a$ be the complete $m$-partite graph with $a$ vertices in each partite set. Let $D(G,k)$ denote the minimum defect in a fractional $k$-coloring of $G$. Complete multipartite fractional-defect conjecture. The minimum defect in a $k$-coloring of $K^{(m)}_a$ is

$$\left(\left\lceil \frac{m}{k}\right\rceil-1\right)a.$$

When $m$ is a multiple of $k$, this value is known to be optimal. The conjecture concerns the remaining values of $m$ and predicts that the natural upper bound remains exact.