Linear almost-spanning monochromatic tight-cycle conjecture

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, with its edges coloured using $r$ colours. A collection of cycles covers $(1-o(1))n$ vertices if the number of uncovered vertices is $o(n)$ as $n$ tends to infinity. Almost-spanning tight-cycle conjecture. For all $r,k\ge 2$, there is a constant $C=C(k)$ such that every $r$-edge-coloured $K_n^{(k)}$ contains at most $Cr$ monochromatic cycles covering $(1-o(1))n$ vertices. The supplied text gives no resolution; the surrounding discussion connects this almost-spanning assertion to the stronger linear partition conjecture.