Multiplicative reduction conjecture for the toroidal covering parameter

Let $n$ be an odd positive integer, and let $\xi(n)$ denote the toroidal covering parameter. Multiplicative reduction conjecture.

$$\xi(n)=\min\left\{\frac{n}{m}\xi(m): m\text{ divides }n\right\}.$$

This conjecture proposes an exact reduction of the odd case to divisors of $n$, complementing the paper's product upper bound. The paper gives no resolution and notes that few values of $\xi(n)$ are known for odd $n$.