We use photometric redshifts and statistical background subtraction to measure stellar mass functions in galaxy group-mass ($4.5-8times10^{13}~mathrm{M}_odot$) haloes at $1<z<1.5$. Groups are selected from COSMOS and SXDF, based on X-ray imaging and sparse spectroscopy. Stellar mass ($M_{mathrm{stellar}}$) functions are computed for quiescent and star-forming galaxies separately, based on their rest-frame $UVJ$ colours. From these we compute the quiescent fraction and quiescent fraction excess (QFE) relative to the field as a function of $M_{mathrm{stellar}}$. QFE increases with $M_{mathrm{stellar}}$, similar to more massive clusters at $1<z<1.5$. This contrasts with the apparent separability of $M_{mathrm{stellar}}$ and environmental factors on galaxy quiescent fractions at $zsim 0$. We then compare our results with higher mass clusters at $1<z<1.5$ and lower redshifts. We find a strong QFE dependence on halo mass at fixed $M_{mathrm{stellar}}$; well fit by a logarithmic slope of $mathrm{d}(mathrm{QFE})/mathrm{d}log (M_{mathrm{halo}}) sim 0.24 pm 0.04$ for all $M_{mathrm{stellar}}$ and redshift bins. This dependence is in remarkably good qualitative agreement with the hydrodynamic simulation BAHAMAS, but contradicts the observed dependence of QFE on $M_{mathrm{stellar}}$. We interpret the results using two toy models: one where a time delay until rapid (instantaneous) quenching begins upon accretion to the main progenitor (no pre-processing) and one where it starts upon first becoming a satellite (pre-processing). Delay times appear to be halo mass dependent, with a significantly stronger dependence required without pre-processing. We conclude that our results support models in which environmental quenching begins in low-mass ($<10^{14}M_odot$) haloes at $z>1$.
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