  The highly-frustrated spin-1/2 quantum Heisenberg model with both nearest
($J_1$) and next-nearest ($J_2$) neighbor exchange interactions is revisited by
using an extended variational space of projected wave functions that are
optimized with state-of-the-art methods. Competition between modulated
valence-bond crystals (VBCs) proposed in the literature and the Dirac spin
liquid (DSL) is investigated. We find that the addition of a {\it small}
ferromagnetic next-nearest-neighbor exchange coupling $|J_2|>0.09 J_1$ leads to
stabilization of a 36-site unit cell VBC, although the DSL remains a local
minimum of the variational parameter landscape. This implies that the VBC is
not trivially connected to the DSL: instead it possesses a non-trivial flux
pattern and large dimerization.
