Conjectural formula for restricted sumsets in cyclic groups

Let $\mathbb{Z}_n$ be the cyclic group of order $n$. For an $m$-subset of $\mathbb{Z}_n$, let $\rho\hat{\;} (\mathbb{Z}_n,m,h)$ denote the minimum size of its restricted $h$-fold sumset. Let $u(n,m,h)$ be the previously defined lower-bound function, and let $w\hat{\;} (n,m,h)$ be the minimum, over divisors $d$ of $n$, of the restricted sumset sizes arising from the two-partial-coset construction $B_d(n,m)$. Bajnok's restricted-sumset formula conjecture. For all $n$, $m$, and $h$,

$$\rho\hat{\;} (\mathbb{Z}_n,m,h)=\min\{u(n,m,h),w\hat{\;} (n,m,h)\}.$$

The formula incorporates the standard coset construction and the exceptional two-partial-coset constructions known to improve it; all known exceptions arise from those cases, but equality is not proved in general.