Sufficiency of admissibility for enclosings at the extremal order

Let $n$ and $m$ be positive integers such that $m=2n-2$, and let $r\geq 2$. Suppose that $r$, $\mu$, and $\lambda$ are positive integers independent of $n$, with $\mu>\lambda$. A decomposition $\mathcal{G}$ of $\lambda K_n$ into $k$ colors is a partition of the edges of $\lambda K_n$ into $k$ color classes. It is $r$-admissible when it satisfies conditions (C1)--(C4) from the paper. An enclosing $2$-edge-connected $r$-factorization of $\mu K_m$ is a decomposition of $\mu K_m$ into $r$-factors, with each color class extending the corresponding color class of $\mathcal{G}$. The conjecture. If $n$ is sufficiently large, then $\mathcal{G}$ can be enclosed in a $2$-edge-connected $r$-factorization of $\mu K_m$ if and only if

$$rk=\mu(m-1),\qquad rm\text{ is even},\qquad\text{and }\mathcal{G}\text{ is }r\text{-admissible}.$$

This proposes that, for sufficiently large $n$, the necessary numerical and admissibility conditions are also sufficient when $m=2n-2$. The paper notes that the conditions are always satisfied when $n$ is large compared with $r$, but gives a counterexample at smaller order, so the asserted large-$n$ sufficiency remains open.