The asymptotic optimal learning-rate conjecture for a binary symmetric cascade

Let $p=q$ denote the common crossover probability of the teacher's and student's binary symmetric channels, and let $D(a\|b)$ denote the binary relative entropy. The optimal learning rate is the best exponential rate at which the student's error probability decays under admissible teaching and learning strategies.

Huleihel et al.'s conjecture. In the regime $p\to 1/2$, the optimal learning rate of the student is

$$D(1/2\|p)(1+o(1));$$

i.e., it agrees with the teacher's learning rate up to first-order error terms.

This conjecture concerns whether the student can asymptotically match the teacher's learning rate when both communication channels have the same noise level near maximal noise. The source attributes it to Huleihel et al.; its resolution is not given here.