We investigate the high-order diagrammatic series in terms of the fully dressed Green function G and the fully dressed screened interaction W by means of the Diagrammatic Monte Carlo method and explore its convergence properties in connection with the recently discovered multivaluedness of the Luttinger-Ward functional for Hubbard-type models. In particular, we find that the bold-GW series diverges well before the branching point of the Luttinger-Ward functional, and identify diagram topologies responsible for divergence. An analytic continuation beyond convergence radius allows us to recover the exact result in the regime where the bold-G-bare-coupling series exhibits misleading convergence. We further explore the controllability of the GW expansion in strongly correlated regimes.