Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$.
That does not hold if the topology considered on $X^*$ is the weak-star topology and the space is non-reflexive. Let $\tau^*_M$ the Mackey topology on $X^*$ with respect to the duality $(X^*,X)$. Then
$X$ is reflexive iff the dual product is strongly$\times\tau^*_M$ (lower)(upper)(semi-)continuous.
My question is:
$\ \ Q:\ $ If $X$ is a Banach non-reflexive space what is the coarsest topology $\tau$ on the dual $X^*$ for which the dual product is strongly$\times\tau$ continuous in $X\times X^*$?
Clearly $\tau$ is weaker than the strong topology on $X^*$ and finer than $\tau^*_M$.