Existence of sets with prescribed numbers of missing sums and differences

Let $j$ and $k$ be nonnegative integers with $k$ even. Prescribed-defect existence conjecture. There exists a positive integer $n$ and a set $S\subset\{0,1,\dots,n-1\}$ such that $0\in S$, $n-1\in S$, and

$$|S+S|=2n-1-j,\qquad |S-S|=2n-1-k.$$

This is posed as an open problem motivated by the preceding limiting-distribution conjecture. The source notes that Hegarty's methods establish it when $j\ge k/2$, while the general case remains open there.