Extremal diameter conjecture for sumsets

Let $A\subseteq\mathbb{Z}$ be a finite set, let $|A|=k$, and write $\operatorname{diam}(A)=\max A-\min A=d$. For $hA=\{a_1+\cdots+a_h:a_i\in A\}$, extremal diameter conjecture. If $h\in\mathbb{N}$ and $k\geq 3$, then

$$|hA|\geq\left|h\bigl([0,k-2]\cup\{d\}\bigr)\right|.$$

The conjecture is presented as an extension of the preceding lemmas and is trivially tight if true. Nathanson gives an explicit computation of the right-hand side, and the conjecture would imply additional gaps in the possible-size set when $h>k+1$; the source reports a related result for $k=3$.