Finite-n peak at four missing differences

Let $P_n^{\mathrm{M}}(k)$ denote the probability that a uniformly random subset $S\subseteq[n]$ has exactly $k$ missing differences, namely

$$P_n^{\mathrm{M}}(k)=\Pr\bigl(2n-1-|S-S|=k\bigr).$$

Peak-at-four conjecture. For all $n\ge 15$ and every $k\ne4$,

$$P_n^{\mathrm{M}}(4)>P_n^{\mathrm{M}}(k).$$

The paper proves that the limiting value at $k=4$ is larger than the other values, and experimental data suggest that $n=15$ already suffices; the conjecture supplies this explicit finite threshold.