The conjecture on factors of independent transversals in $[k,n,1]$-graphs

Let a $[k,n,1]$-graph be a $k$-partite graph with parts of size $n$ and maximum degree at most $1$, and let $f(k)$ be the smallest integer $n_0$ such that every $[k,n,1]$-graph has a factor of independent transversals whenever $n\ge n_0$. Here, a factor of independent transversals is a partition of the vertices into independent sets, each containing exactly one vertex from every part.

The conjecture on $f(k)$. For every $k\ge 2$,

$$\nf(k)=\begin{cases}k,&\text{if }k\text{ is even},\\ k+1,&\text{if }k\text{ is odd}.\end{cases}$$

The lower bounds arise from explicit constructions, including Catlin's construction when $k\ge 3$ is odd. The conjecture is trivial for $k=2$, known for $k=3$ and $k=4$, and is otherwise wide open; existing asymptotic results for related modified Fischer conjectures do not imply this precise bound.