Lemma: Compact signalling query

The strength is the attained maximum

and belongs to $[0,1]$.

Proof. Total variation of finite laws equals the maximum probability difference over outcome subsets. Coarse-graining such a subset gives one element $B\in {\mathsf{B}}_b$.

Conversely, every such $B$ extends to a binary instrument by adding

This is positive and makes the sum deterministic. The binary total variation equals the displayed absolute probability difference.

All tester sets are closed and consist of positive matrices of trace at most two. They are compact. The continuous objective therefore attains its maximum, and the binary probability difference is at most one.