Given $(X,s)$ a (real) barreled locally convex space (that is, every closed convex and absorbing set in $(X,s)$ is a neighborhood of the origin), is there a (strictly) finer, non-barreled linear locally convex topology $\tau$ on $X$ that is compatible with the duality $(X,X^*)$?
In other words $s\preceq\tau$, $(X,s)^*=(X,\tau)^*$, and $\tau$ is not barreled.