The dimension conjecture for the losing set of lower digit frequencies

Let $D^-_{c}$ denote the set of real numbers whose lower frequency of the digit $0$ is greater than $c$, and let $D(D^-_{c})$ denote its set of Schmidt-game parameters $(\alpha, \beta)$ for which it is winning. The dimension conjecture. We believe that

$$D(D^-_{c}) = \{ (\alpha, \beta): \, \log(\alpha)/\log(\alpha\beta) > c \}.$$

This predicts the exact parameter region in which the lower digit-frequency set is winning. The supplied text does not state any resolution of the conjecture.