Information-distillation lower-bound tightness conjecture

Let $\mathcal{X}$ be a finite alphabet with $|\mathcal{X}|>2$. For an information-distillation function $\mathrm{ID}_M(|\mathcal{X}|,\beta)$, where $M$ is the number of quantization levels and $\beta$ is the mutual-information scale, the conjecture concerns maintaining a fraction $\eta(|\mathcal{X}|)$ of that scale as $\beta$ tends to zero. Information-distillation lower-bound tightness conjecture. For any $|\mathcal{X}|>2$, there exist $\eta(|\mathcal{X}|)$, $\beta(|\mathcal{X}|)>0$, and $c(|\mathcal{X}|)>0$ such that

$$\frac{|\mathcal{X}|-2}{|\mathcal{X}|-1}<\eta(|\mathcal{X}|)<1,$$

and, for all $0<\beta<\beta(|\mathcal{X}|)$ and $M<c(|\mathcal{X}|)(\log(1/\beta))^{|\mathcal{X}|-1}$,

$$\mathrm{ID}_M(|\mathcal{X}|,\beta)<\eta(|\mathcal{X}|)\cdot\beta.$$

The preceding bounds establish tight quantization-level growth for fractions below $1/(|\mathcal{X}|-1)$, while the conjecture would extend a matching lower-bound characterization to a fraction strictly larger than $(|\mathcal{X}|-2)/(|\mathcal{X}|-1)$ and arbitrarily close to $1$. The statement remains unproved in the source.