  We apply a random-plaquette $J_1$-$J_2$ model on the square lattice to
capture the physics of a series of spin-$1/2$ Heisenberg antiferromagnet
compounds Sr$_2$CuTe$_{1-x}$W$_x$O$_6$. With the input of experimentally
relevant coupling strengths, our exact diagonalization (ED) study probes the
ground state properties beyond previous linear spin-wave approach. An
intermediate range of $x \in [0.08, 0.55]$ is identified for a nonmagnetic
phase without the long-range N\'eel or stripe order. The absence of both
valence-bond-glass order and spin-glass non-ergodic dynamics renders its nature
intriguing. Deep inside this phase around $x = 0.3$, we observe signatures
potentially linked to randomness-induced short-range spin-liquid-like (SLL)
states, including close to zero spin-freezing parameter, vanishing spin-spin
correlation beyond nearest neighbors, almost uniform static spin structure
factor, as well as a broad tail in the dynamical spin structure factor. The
nonmagnetic phase also features multipartite entanglement in the ground state
witnessed by quantum Fisher information (QFI), which exhibits universal scaling
behaviors at quantum critical points.
