Possible-size interval conjecture for sumsets

Let $\mathcal{R}(h,k)$ denote the set of possible cardinalities $|hA|$ for $k$-element sets $A$ of integers, and let $\Delta_{h,k}$ be the exceptional set defined in the paper. Possible-size interval conjecture. For every $h\in\mathbb{N}$ and every $k>h$,

$$\mathcal{R}(h,k)=\left[hk-h+1,\binom{h+k-1}{h}\right]\setminus\Delta_{h,k}.$$

Theorem~ proves this for each fixed $h$ and all sufficiently large $k$, while numerical experiments suggest that the threshold $k>h$ may be best possible; the claim can fail when $k\leq h$.