Let $V$ be a Banach space and $B(V)$ be the bounded operators on $V$. Now, the space $B(V)$ has a norm, the strong topology (which is the topology of pointwise convergence) and it has a stronger topology called ultrastrong or $\sigma$-strong topology. It is known that these two topologies coincide on norm-bounded sets. I want to know if the $\sigma$-strong topology is the strongest topology which coincides with the strong topology (respectively itself) on bounded sets.

In fact, I am interested in the special case where $V$ is a C*-algebra. In this case, $B(V)$ gets replaced by the multiplier algebra. Is in this setting the $\sigma$-strong-$*$ topology the strongest topology which coincides with itself on bounded sets?

A positive first result would imply a positive second one but I fear that the first one fails while the second one might hold true.