Exact unconditional signalling

For distinct $a,b\in \mathsf{J}$, let $c$ be the remaining party. The sender uses deterministic testers $S,T\in {\mathsf{C}}_a$. The receiver uses an instrument $\{B_z\}$, and $c$ uses $M_c\in {\mathsf{C}}_c$. Define

where $d_{\mathrm{TV}}(P,Q)=\frac{1}{2}{\sum }_z|P(z)-Q(z)|$. Tensor factors always occupy their named slots.