Precipitation extremes produced by convection have been found to intensify with near-surface temperatures at a Clausius-Clapeyron rate of $6$ to $7\%$ K$^{-1}$ in simulations of radiative-convective equilibrium (RCE). However, these idealized simulations are typically performed over an ocean surface with a high near-surface relative humidity (RH) that stays roughly constant with warming. Over land, near-surface RH is lower than over ocean and is projected to decrease by global climate models. Here, we investigate the dependence of precipitation extremes on near-surface RH in convection-resolving simulations of RCE. We reduce near-surface RH by increasing surface evaporative resistance while holding free-tropospheric temperatures fixed by increasing surface temperature. This ``top-down'' approach produces an RCE state with a deeper, drier boundary layer, which weakens convective precipitation extremes in three distinct ways. First, the lifted condensation level is higher, leading to a small thermodynamic weakening of precipitation extremes. Second, the higher lifted condensation level also reduces positive buoyancy in the lower troposphere, leading to a dynamic weakening of precipitation extremes. Third, precipitation re-evaporates more readily when falling through a deeper, drier boundary layer, leading to a substantial decrease in precipitation efficiency. These three effects all follow from changes in near-surface relative humidity and are physically distinct from the mechanism that underpins the Clausius-Clapeyron scaling rate. Overall, our results suggest that changes in relative humidity must be taken into account when seeking to understand and predict changes in convective precipitation extremes over land.