Pasotti–Pellegrini's odd-order near-factor conjecture

Let $K_v$ be the complete graph on $v$ vertices, where $v=2n+1$ is odd, and define edge-length by

$$\ell(x,y)=\min(|x-y|,v-|x-y|).$$

A near $1$-factor $F$ is a spanning subgraph of $K_v$ with one isolated vertex and a matching on the remaining vertices. Let $L$ be a list of $n$ positive integers not exceeding $n$, and call it admissible when, for every divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ is at most $\frac{v-d}{2}$.

Pasotti–Pellegrini conjecture. There exists a near $1$-factor $F$ of $K_v$ with $\ell(F)=L$ if and only if $L$ is admissible.

The source reports partial results but states that this conjecture remains widely open. It generalizes Meszka's problem from prime order to arbitrary odd order.