Mohan–Pandey conjecture on near-minimal restricted h-fold sumsets

Let $k$ be a large positive integer and let $h$ be a positive integer with

$$2\leq h\leq k-2.$$

Let $A$ be a finite set of $k$ nonnegative integers with $\min(A)=0$ and $d(A)=1$, where $d(A)$ is the greatest common divisor of all differences between elements of $A$. Write $h^{\wedge}A$ for the restricted $h$-fold sumset, consisting of sums of $h$ distinct elements of $A$.

Mohan–Pandey conjecture. The following implications hold:

(a) If $\left|h^{\wedge}A\right|=hk-h^2+2$, then $A\subset[0,k]$.

(b) If $\left|h^{\wedge}A\right|=hk-h^2+3$, then $A\subset[0,k+1]$.

(c) If $\left|h^{\wedge}A\right|=hk-h^2+4$, then $A\subset[0,k+2]$.

The conjecture extends the cited classifications of restricted double, triple, and quadruple sumsets to arbitrary positive $h$ in the stated range. The source describes $k$ only as large and gives no resolution.