Pach–Saghafian–Schnider conjecture on decompositions of cliques into k-star-forests

A $k$-star-forest is a star-forest with at most $k$ connected components. Let $F_k(n)$ be the minimum integer such that the complete graph on $n$ vertices can be decomposed into $F_k(n)$ $k$-star-forests.

Pach–Saghafian–Schnider conjecture. For any $n\ge k\ge 2$,

$$F_k(n)\ge \left\lceil\frac{(k+1)n}{2k}\right\rceil.$$

This lower bound asserts that the construction obtained from the broken double-star decomposition together with matchings of size at most $k$ is best possible for decomposing cliques into $k$-star-forests. The statement is presented as a conjecture in the source; no resolution is given here.