The monochromatic tree-cover averaging conjecture

Let $t$ be a positive integer, let $s$ satisfy $1\le s\le t-1$, and let $K_n$ be a complete graph with its edges colored using $t$ colors. A monochromatic tree is a tree all of whose edges have one color. The averaging conjecture. At least

$$\frac{ns}{t-1}$$

vertices can be covered by the vertices of at most $s$ monochromatic trees. This follows from the preceding Ryser-type conjecture by an averaging argument, and is therefore open in the same generality.