  Inferring high-fidelity constraints on the spatial curvature parameter,
$\Omega_{\rm K}$, under as few assumptions as possible, is of fundamental
importance in cosmology. We propose a method to non-parametrically infer
$\Omega_{\rm K}$ from late-Universe probes alone. Using Gaussian Processes (GP)
to reconstruct the expansion history, we combine Cosmic Chronometers (CC) and
Type Ia Supernovae (SNe~Ia) data to infer constraints on curvature,
marginalized over the expansion history, calibration of the CC and SNe~Ia data,
and the GP hyper-parameters. The obtained constraints on $\Omega_{\rm K}$ are
free from parametric model assumptions for the expansion history, and are
insensitive to the overall calibration of both the CC and SNe~Ia data (being
sensitive only to relative distances and expansion rates). Applying this method
to \textit{Pantheon} SNe~Ia and the latest compilation of CCs, we find
$\Omega_{\rm K} = -0.03 \pm 0.26$, consistent with spatial flatness at the
$\mathcal{O}(10^{-1})$ level, and independent of any early-Universe probes.
Applying our methodology to future Baryon Acoustic Oscillations and SNe~Ia data
from upcoming Stage IV surveys, we forecast the ability to constrain
$\Omega_{\rm K}$ at the $\mathcal{O}(10^{-2})$ level.
