Linear independence of a hypercyclic orbit for semigroups

A question of Conejero and Peris from 2005 asks the following: if {T(t)}t≥0 is a strongly continuous semigroup of operators acting on a Banach space, is the orbit {T(t)x:t≥0} of a hypercyclic vector x always linearly independent? We give a positive answer to this question for complex Banach spaces under the assumption that the spectrum σ(A) of the generator A of the semigroup is nonempty.