  The spin-$1/2$ $J_1-J_2$ Heisenberg model on the square lattice represents
one of the simplest examples in which the effets of magnetic interactions may
suppress magnetic order, eventually leading to a pure quantum phase with no
local order parameters. This model has been extensively studied in the last
three decades, with conflicting results. Here, by using Gutzwiller-projected
wave functions and recently developed methods to assess the low-energy
spectrum, we show the existence of a level crossing between the lowest-energy
triplet and singlet excitations for $J_2/J_1 \approx 0.54$. This fact supports
the existence of a phase transition between a gapless spin liquid (which is
stable for $0.48 \lesssim J_2/J_1 \lesssim 0.54$) and a valence-bond solid (for
$0.54 \lesssim J_2/J_1 \lesssim 0.6$), even though no clear sign of dimer order
is visible in the correlations functions. These results, which confirm recent
density-matrix renormalization calculations on cylindrical clusters [L. Wang
and A.W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)] reconcile the
contraddicting results obtained within different approaches over the years.
