  The dearth of planets with sizes around 1.8 $\mathrm{R_\oplus}$ is a key
demographic feature discovered by the $Kepler$ mission. Two theories have
emerged as potential explanations for this valley: photoevaporation and
core-powered mass-loss. However, Rogers et al. (2021) shows that
differentiating between the two theories is possible using the
three-dimensional parameter space of planet radius, incident flux, and stellar
mass. We use homogeneously-derived stellar and planetary parameters to measure
the $Kepler$ exoplanet radius gap in this three-dimensional space. We compute
the slope of the gap as a function of incident flux at constant stellar mass
($\alpha$ $\equiv$ $\left(\partial \log R_{\mathrm{gap}} / \partial \log S
\right)_{M_\star}$) and the slope of the gap as a function of stellar mass at
constant incident flux ($\beta$ $\equiv$ $\left(\partial \log R_{\mathrm{gap}}
/ \partial \log M_\star \right)_{S}$) and find $\alpha$ =
0.069$^{+0.019}_{-0.023}$ and $\beta$ = $-$0.046$^{+0.125}_{-0.117}$. Given
that Rogers et al. (2021) shows that core-powered mass-loss predicts $\alpha$
$\approx$ 0.08 and $\beta$ $\approx$ 0.00 while photoevaporation predicts
$\alpha$ $\approx$ 0.12 and $\beta$ $\approx$ --0.17, our measurements are more
consistent with core-powered mass-loss than photoevaporation. However, we
caution that different gap-determination methods can produce systematic offsets
in both $\alpha$ and $\beta$; therefore, we motivate a comprehensive
re-analysis of $Kepler$ light curves with modern, updated priors on
eccentricity and mean stellar density to improve both the accuracy and
precision of planet radii and subsequent measurements of the gap.
