The odd-order near-factor characterization conjecture

Let $v=2n+1$ be an odd integer, let $K_v$ be the complete graph on $\{0,1,\ldots,v-1\}$, and let $L$ be a list of $n$ positive integers not exceeding $n$. A near $1$-factor $F$ is a set of $n$ pairwise disjoint edges, and $\ell(F)$ is its list of edge-lengths. The near-factor characterization conjecture. There exists a near $1$-factor $F$ of $K_v$ such that $\ell(F)=L$ if and only if, for any divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ does not exceed $\frac{v-d}{2}$. This is presented as a generalization of Bacher's problem for odd order. The supplied text gives no resolution.