The generalized dihedral MSTD–MDTS conjecture

Let $G$ be an abelian group containing an element of order at least $3$, and let $D=\mathbb Z_2\ltimes G$ be the generalized dihedral group in which the nonidentity element of $\mathbb Z_2$ acts on $G$ by inversion. A subset is MSTD if its sumset has greater cardinality than its difference set, and MDTS if its difference set has greater cardinality than its sumset.

Generalized dihedral MSTD–MDTS conjecture. There are more MSTD subsets of $D$ than MDTS subsets of $D$.

The ordinary dihedral conjecture is the special case $G=\mathbb Z_n$. The supplied text gives no resolution of the generalized assertion.