The inverse-exponent conjecture for direct factors of cyclic groups

Let $k$, $\omega$, and $n$ be positive integers, and let $\mathbb{Z}_{\omega}$ be a finite cyclic group of order $\omega$. Suppose

$$A=[0,n-k-1]\cup\{i_0,i_1,\ldots,i_{k-1}\}\subseteq\mathbb{Z}_{\omega},\qquad |A|=n\geq 2k+1.$$

A subset $A$ is a direct factor of $\mathbb{Z}_{\omega}$ if it occurs as one factor in a direct factorization of that cyclic group. Inverse-exponent conjecture. If $A$ is a direct factor of $\mathbb{Z}_{\omega}$, then

$$\{i_0,i_1,\ldots,i_{k-1}\}\equiv\{n-k,n-k+1,\ldots,n-1\}\pmod n.$$

The conjecture is the cyclic-group factorization counterpart of the multiplier-set conjecture above; the source notes that the corresponding assertion is known for $k\leq 2$, while the general case remains open.