  When spin-orbit-entangled $d$-electrons reside on a honeycomb lattice, rich
quantum states are anticipated to emerge, as exemplified by the $d^5$ Kitaev
materials. Distinct yet equally intriguing physics may be realized with a
$d$-electron count other than $d^5$. We found that the layered ruthenate
Ag$_3$LiRu$_2$O$_6$ with $d^4$ Ru$^{4+}$ ions at ambient pressure forms a
honeycomb lattice of spin-orbit-entangled singlets, which is a playground for
frustrated excitonic magnetism. Under pressure, the singlet state does not
develop the expected excitonic magnetism but experiences two successive
transitions to other nonmagnetic phases, first to an intermediate phase with
moderate distortion of honeycomb lattice, and eventually to a high-pressure
phase with very short Ru-Ru dimer bonds. While the strong dimerization in the
high-pressure phase originates from a molecular orbital formation as in the
sister compound Li$_2$RuO$_3$, the intermediate phase represents a
spin-orbit-coupled $J$-dimer state which is stabilized by the admixture of
upper-lying $J_{\rm eff} = 1$-derived states. We argue that the $J$-dimer state
is induced by a pseudo-Jahn-Teller effect associated with the low-lying
spin-orbital excited states and is unique to spin-orbit-entangled $d^4$
systems. The discovery of competing singlet phases demonstrates rich
spin-orbital physics of $d^4$ honeycomb compounds and paves the way for
realization of unconventional magnetism.
