Separation-number conjecture for complete graphs

Let $K_n$ be the complete graph, and let $\mathsf{sep}(K_n,a,b)$ denote its separation threshold for $a$-list assignments and $b$-fold colorings. Let $n\geq 4$, let $a,b$ be the list and coloring parameters, and let $p$ satisfy

$$2\leq p\leq n-2,\qquad pb\leq a<(p+1)b.$$

Separation-number conjecture. Under these conditions,

$$\mathsf{sep}(K_n,a,b)=\left\lceil\frac{2pa-p(p+1)b}{n-1}\right\rceil+\epsilon,\qquad \epsilon\in\{-1,0\}.$$

The conjecture combines the partial results preceding it and predicts the separation threshold across the intermediate ranges of $a/b$. The supplied text presents it as unresolved; the correction term $\epsilon$ may depend on $a,b,n$.