Let $X$ be a locally compact Hausdorff space. Consider the dual pair $\langle C_b(X), M_R(X) \rangle$ between continuous bounded functions and signed Radon measures with bounded variation on $X$. A compatible topology on $C_b(X)$ is Buck's strict topology. If $X$ is not compact then the norm topology on $C_b(X)$ is not compatible with the duality: the norm dual of $C_b(X)$ is $M_R(\beta X)$.
Does the strong topology $\beta(C_b(X), M_R(X))$ coincide with the norm topology? In other words, does it hold $\beta(C_b(X), M_R(X)) = \beta(C_b(X), M_R(\beta X))$? (If not, is there a known description of the $\beta(C_b(X), Rca(X))$-dual of $C_b(X)$ in terms of a subspace of $M_R(\beta X)$?)