The local characterization conjecture for multiplier sets and cyclic direct factors

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

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

For a prime $p$ dividing $|G|$, write $\operatorname{ord}_p(m)$ for the multiplicative order of $m$ modulo $p$. Also let $\mathbb{Z}_{\omega}$ be a cyclic group of order $\omega$, and let $A=[0,n-k-1]\cup\{i_0,i_1,\ldots,i_{k-1}\}\subseteq\mathbb{Z}_{\omega}$ with $|A|=n$. Local characterization conjecture. The following assertions hold:

(i) $M$ splits an abelian group $G$ with $|M|=n$ if and only if, for every prime $p\mid |G|$,

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

and $n\mid\operatorname{ord}_p(m)$.

(ii) $A$ is a direct factor of $\mathbb{Z}_{\omega}$ if and only if

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

and $n\mid\omega$.

The source explains that part (i) and part (ii) are equivalent to the two preceding conjectures, respectively, and that they are known for $k\leq 2$; the general characterization remains open.