  The quest for entangled spin excitations has stimulated intense research on
frustrated magnetic systems. For almost two decades, the triangular-lattice
Mott insulator $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ has been the hottest
candidate for a $gapless$ quantum spin liquid with itinerant spinons. Very
recently, however, this scenario was overturned as electron-spin-resonance
(ESR) studies unveiled a spin gap, calling for reevaluation of the magnetic
ground state. Here we achieve a precise mapping of this spin-gapped phase
through the Mott transition by ultrahigh-resolution strain tuning. Our
transport experiments reveal a reentrance of charge localization below
$T^{\star}=6$ K associated with a gap size of 30-50 K. The negative slope of
the insulator-metal boundary, $dT^{\star}/dp<0$, evidences the low-entropy
nature of the spin-singlet ground state. By tuning the enigmatic '6 K anomaly'
through the phase diagram of $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$, we identify
it as the transition to a valence-bond-solid phase, with typical magnetic and
structural fingerprints, that persists at $T\rightarrow 0$ until unconventional
superconductivity and metallic transport proliferate.
