Quantum spin liquids (QSLs) are exotic phases of matter exhibiting long-range entanglement and supporting emergent gauge fields. A vigorous search for experimental realizations of these states has identified several materials with properties hinting at QSL physics. A key issue in understanding these QSL candidates is often the interplay of weak disorder of the crystal structure with the spin liquid state. It has recently been pointed out that in at least one important class of candidate QSLs - pyrochlore magnets based on non-Kramers ions such as Pr$^{3+}$ or Tb$^{3+}$- structural disorder can actually promote a $U(1)$ QSL ground state. Here we set this proposal on a quantitative footing by analyzing the stability of the QSL state in the minimal model for these systems: a random transverse field Ising model. We consider two kinds of instability, which are relevant in different limits of the phase diagram: condensation of spinons and confinement of the $U(1)$ gauge fields. Having obtained stability bounds on the QSL state we apply our results directly to the disordered candidate QSL Pr$_2$Zr$_2$O$_7$. We find that the available data for currently studied samples of Pr$_2$Zr$_2$O$_7$ is most consistent with it a ground state outside the spin liquid regime, in a paramagnetic phase with quadrupole moments near saturation due to the influence of structural disorder.
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