Bhanja–Kom–Pandey lower-bound and inverse conjecture for positive restricted signed sumsets

Let $A$ be a set of $k\geq 4$ positive integers, and let $h$ be an integer with $3\leq h\leq k-1$. The restricted signed $h$-fold sumset is denoted by $h^{\wedge}_{\pm}A$. Bhanja–Kom–Pandey's conjecture. One should have

$$\left|h^{\wedge}_{\pm}A\right|\geq 2hk-h^2+1.$$

This lower bound is best possible. Moreover, if equality holds, then

$$A=d\ast\{1,3,\ldots,2k-1\}$$

for some positive integer $d$. The conjecture gives the expected lower bound and equality case for positive-integer restricted signed sumsets; the source states that the case $h=3$ was proved, while the general range remains conjectural.