The inverse-exponent conjecture for multiplier sets

Let $m$, $n$, and $k$ be positive integers with $m\geq 2$, and suppose

$$M=\{1,m,m^2,\ldots,m^{n-k-1},m^{i_0},m^{i_1},\ldots,m^{i_{k-1}}\}.$$

A multiplier set $M$ splits an abelian group $G$ if there is a set $S\subseteq G$ such that every nonzero element of $G$ has a unique representation $ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. Inverse-exponent conjecture. If $M$ splits an abelian group $G$ and $|M|=n\geq 2k+1$, then

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

This generalizes the preceding result for $k=1$; the source notes that the conjecture is known for $k\leq 2$, while the general case remains open.