  We study the parameterized complexity of two classic problems on directed
graphs: Hamiltonian Cycle and its generalization {\sc Longest Cycle}. Since
2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by
directed treewidth [Lampis et al., ISSAC'08]. By now, the question of whether
it is FPT parameterized by the directed feedback vertex set (DFVS) number has
become a longstanding open problem. In particular, the DFVS number is the
largest natural directed width measure studied in the literature. In this
paper, we provide a negative answer to the question, showing that even for the
DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain
that Longest Cycle is W[1]-hard on directed graphs when parameterized
multiplicatively above girth, in contrast to the undirected case. This resolves
an open question posed by Fomin et al. [ACM ToCT'21] and Gutin and Mnich
[arXiv:2207.12278]. Our hardness results apply to the path versions of the
problems as well. On the positive side, we show that Longest Path parameterized
multiplicatively above girth} belongs to the class XP.
