The dense digit-pair conjecture for kernel-expandability

Let $b$ be the base and let $D_1,D_2\subseteq\{0,1,\ldots,b-1\}$ be digit sets, with $\#D_1$ and $\#D_2$ denoting their cardinalities. A pair $(D_1,D_2)$ is called degenerate or Kernel-expandable according to the definitions in the paper. Dense digit-pair conjecture. There is a number $0<r<1$ close to $1$ such that, for all sufficiently large $b$, if

$$\frac{\#D_1}{b}>r\qquad\text{and}\qquad\frac{\#D_2}{b}>r,$$

then $(D_1,D_2)$ is either degenerate or Kernel-expandable. The conjecture predicts that forbidding only a sufficiently small proportion of digits forces this dichotomy; its resolution is not given in the source.