The fixed-size MSTD–MDTS conjecture for dihedral groups

Let $D_{2n}$ be the dihedral group of order $2n$. For a positive integer $m$, let $\mathcal S_{2n,m}$ denote the collection of subsets of $D_{2n}$ of size $m$. 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.

Fixed-size MSTD–MDTS conjecture. For every integer $n\geq 3$ and every $m\leq 2n$, the collection $\mathcal S_{2n,m}$ has at least as many MSTD sets as MDTS sets.

The source reports that this conjecture is known for $m=2$, $m=3$, and $m>n$; the general fixed-size assertion remains open in the supplied text.