The strict causal penalty conjecture for optimal teaching and learning

Let $p$ and $q$ be the crossover probabilities of the teacher's and student's binary symmetric channels, respectively, with

$$0<p,q<1/2.$$

Let $D(a\|b)$ denote the binary relative entropy, and let the optimal learning rate be the supremum over all causal joint teaching and learning strategies.

Causal penalty conjecture. The optimal learning rate of the student is strictly less than

$$\min\bigl(D(1/2\|p),D(1/2\|q)\bigr).$$

The conjecture asserts that causal teaching and learning impose a strict penalty in the binary setting, despite the data-processing upper bound. The source notes that this penalty need not occur in the Gaussian setting and that the corresponding non-causal upper bound may be achieved; determining optimal causal joint strategies remains open.