Grinblat's conjecture on rainbow matchings in $(n,v)$-multigraphs

An $(n,v)$-multigraph is an $n$-edge-coloured multigraph in which the edges of each colour span a disjoint union of non-trivial cliques whose total number of vertices is at least $v$. Let $v(n)$ be the minimum integer $v$ such that every $(n,v)$-multigraph contains a rainbow matching of size $n$. Grinblat's conjecture. For all $n\geq 4$,

$$v(n)=3n-2.$$

This problem generalises rainbow matching questions such as the bipartite Aharoni–Berger conjecture and arose from Grinblat's study of algebras of sets; the conjecture remains open here.