Divot classification for missing differences

Let $S$ be chosen uniformly at random from $[n]=\{0,1,\dots,n-1\}$. Write

$$P_n^{\mathrm{H}}(k)=\Pr\bigl(|S-S|=k\bigr)$$

for the probability that the diffset has size $k$, and call $k$ a divot when its probability is lower than the neighboring values in the distribution. For $k\ll n$, the typical diffset size is $k(k-1)+1$.

Divot classification conjecture. For every $k>1$, $k(k-1)+3$ is a divot of $P_n^{\mathrm{H}}$ for sufficiently large $n$. Furthermore, these are the only divots.

The conjecture predicts the complete pattern of divots in the distribution of diffset sizes when the selected-set size is small relative to $n$. The source gives no resolution status.