No free lunch for continuity of stochastic convolutions

We construct compact, exponentially stable $C_0$-semigroups $S$ on Hilbert spaces $X$ such that, for every $T>0$, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}β_s $ is unbounded on $[0,T]$ with positive probability for some predictable $g\in L^\infty(Ω;L^2(0,T;X))$, where $β$ is