The equal-sum partition conjecture for cyclic groups of order $2^n$

Let $\mathbb{Z}_m$ be a cyclic group of order $m=2^n$. Let $t$ and $m_1,\ldots,m_t$ be positive integers satisfying

$$\sum_{i=1}^t m_i=m-1,$$

with $m_1\geq 1$ and $m_i\geq 2$ for $2\leq i\leq t$. Equal-sum partition conjecture. There exists $\mu\in\mathbb{Z}_m$ such that the nonzero elements of $\mathbb{Z}_m$ can be partitioned into disjoint subsets $S_1,\ldots,S_t$ with $|S_i|=m_i$ and

$$\sum_{s\in S_i}s=\mu$$

for every $i$, $1\leq i\leq t$. This is presented as an equivalent formulation of the complete multipartite distance-magic conjecture, whose status is not separately resolved in the source.