Let $X$ be a completely regular Hausdorff space, $C_b(X)$ the Banach space of bounded continuous function and $M(X) \subseteq M(\beta X) = C(\beta X)' = C_b(X)'$ the spaces of Radon measures on $X$. For the strong topologies it then holds

$$ \beta(C_b(X), M(X)) \subseteq \beta(C_b(X), M(\beta X)) = \lVert \cdot \rVert\textrm{-topology}. $$

Is it true that $\beta(C_b(X), M(X)) = \lVert \cdot \rVert\textrm{-topology}$ (or at least if $X$ is locally compact)? One has to show that every bounded set in $M(\beta X)$ is contained in the closure (or bipolar) of a bounded set of $M(X)$.

If not, what is the $\beta(C_b(X), M(X))$-dual in terms of measures?

(This is a question I posted in MSE a few days ago, which showed low interest there.)