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The highly-frustrated spin-1/2 quantum Heisenberg model with both nearest($J_1$) and next-nearest ($J_2$) neighbor exchange interactions is revisited byusing an extended variational space of projected wave functions that areoptimized with state-of-the-art methods. Competition between modulatedvalence-bond crystals (VBCs) proposed in the literature and the Dirac spinliquid (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 tostabilization of a 36-site unit cell VBC, although the DSL remains a localminimum of the variational parameter landscape. This implies that the VBC isnot trivially connected to the DSL: instead it possesses a non-trivial fluxpattern and large dimerization.

Valence-bond crystal in the extended kagome spin-1/2 quantum Heisenberg antiferromagnet: A variational Monte Carlo approach