Positive limiting proportions for prescribed missing sums and differences

For $n\ge 1$, let $S$ range over subsets of $\{0,1,\dots,n-1\}$. For nonnegative integers $j$ and $k$, define

$$\rho_{j,k}:=\lim_{n\to\infty}2^{-n}\#\{S\subset\{0,1,\dots,n-1\}:|S+S|=2n-1-j,\ |S-S|=2n-1-k\},$$

assuming the limit exists. Since $S-S$ is symmetric about $0$, only even $k$ can occur. The limiting-distribution conjecture. For every pair of nonnegative integers $j$ and $k$ with $k$ even, $\rho_{j,k}$ exists and is positive; moreover,

$$\sum_{j=0}^{\infty}\sum_{\substack{k=0\k\text{ even}}}^{\infty}\rho_{j,k}=1.$$

This refines the conjecture on the three aggregate limiting proportions by recording the exact numbers of missing sums and differences. The normalization asserts that these limiting frequencies account for all subsets asymptotically; the source also notes that the claim is known when $j\ge k/2$.