Conjectural formula for unrestricted-length restricted sumsets in cyclic groups

Let $n$ be a positive integer and let $m\leq n$. For an $m$-subset $A$ of $\mathbb{Z}_n$, write $\Sigma A=\bigcup_{h=0}^{\infty}h\hat{\;}A$ for its restricted sumset over all numbers of terms, and let $\rho\hat{\;} (\mathbb{Z}_n,m,\mathbb{N}_0)$ be the minimum possible size of $\Sigma A$. Let $u(n,m,\mathbb{N}_0)$ be the divisor-based function defined from the constructions $C_d(n,m)$. Unrestricted-length restricted-sumset formula conjecture. For all positive integers $n$ and $m\leq n$,

$$\rho\hat{\;} (\mathbb{Z}_n,m,\mathbb{N}_0)=u(n,m,\mathbb{N}_0).$$

The construction gives the upper bound, and numerical experimentation in the source supports equality; a proof is not supplied.