Brownian moving averages have conditional full support

We prove that any Brownian moving average $$X_t={\int }_{-\mathrm{\infty }}^t(f(s-t)-f(s))d{B}_s,t\ge 0,$$ satisfies the conditional full support condition introduced by Guasoni, R'{a}sonyi and Schachermayer [Ann. Appl. Probab. 18 (2008) 491–520].