A classical result states that: If $X$ is a Banach space then every multi-valued monotone operator $T:X\to 2^{X^*}$ is locally bounded on $\operatorname*{int}D(T)$ (the interior of its domain).
I wonder whether this result still holds in a general locally convex space $X$. Here $X^*$ is the topological dual of $X$ and the weakest possible boundedness, namely, the weak-star boundedness of subsets of $X^*$.
More precisely, $T:X\to 2^{X^*}$ is locally weak-star bounded on $\operatorname*{int}D(T)$ if, for every $x_0\in\operatorname*{int}D(T)$, there is $V$ a neighborhood of $x_0$ such that $T(V)$ is weak-star bounded in $X^*$.