  We prove that any bounded linear operator on $L_p[0,1]$ for $1\leq p<\infty$,
commuting with the Volterra operator $V$, is not weakly supercyclic, which
answers affirmatively a question raised by L\'eon-Saavedra and Piqueras-Lerena.
It is achieved by providing an algebraic flavored condition on an operator
which prevents it from being weakly supercyclic and is satisfied for any
operator commuting with $V$.
