The non-uniform 1-factorability congruence conjecture

Let $cmathcal{L}$ be a set of distinct positive integers whose largest element is $k$. The non-uniform set system $cbinom{[n]}{cmathcal{L}}$ consists of the subsets of $[n]$ whose sizes belong to $cmathcal{L}$, and it is 1-factorable when its edges can be partitioned into 1-factors. The non-uniform 1-factorability congruence conjecture. Depending on the choice of $cmathcal{L}$, exactly one of the following statements must be true:

$$\text{(i) For sufficiently large }n,\ \binom{[n]}{\mathcal{L}}\text{ is 1-factorable if and only if }n\equiv 0\pmod{k};$$

or

$$\text{(ii) For sufficiently large }n,\ \binom{[n]}{\mathcal{L}}\text{ is 1-factorable if and only if }n\equiv 0,-1\pmod{k}.$$

The preceding results establish the relevant factorability results in the cases treated, while the conjecture seeks a complete classification for every choice of $cmathcal{L}$; in particular, the behavior when $k-1\in\mathcal{L}$ can differ between the two scenarios.