  It has so far proven impossible to reproduce all aspects of the solar plage
chromosphere in quasi-realistic numerical models. The magnetic field
configuration in the lower atmosphere is one of the few free parameters in such
simulations. The literature only offers proxy-based estimates of the field
strength, as it is difficult to obtain observational constraints in this
region. Sufficiently sensitive spectro-polarimetric measurements require a high
signal-to-noise ratio, spectral resolution, and cadence, which are at the limit
of current capabilities. We use critically sampled spectro-polarimetric
observations of the \cair line obtained with the CRISP instrument of the
Swedish 1-m Solar Telescope to study the strength and inclination of the
chromospheric magnetic field of a plage region. This will provide direct
physics-based estimates of these values, which could aid modelers to put
constraints on plage models. We increased the signal-to-noise ratio of the data
by applying several methods including deep learning and PCA. We estimated the
noise level to be $1\cdot10^{-3} I_c$. We then used STiC, a non-local
thermodynamic equilibrium (NLTE) inversion code to infer the atmospheric
structure and magnetic field pixel by pixel. We are able to infer the magnetic
field strength and inclination for a plage region and for fibrils in the
surrounding canopy. In the plage we report an absolute field strength of $|B|
=440 \pm 90$ G, with an inclination of $10^\circ \pm 16^\circ$ with respect to
the local vertical. This value for $|B|$ is roughly double of what was reported
previously, while the inclination matches previous studies done in the
photosphere. In the fibrillar region we found $|B| = 300 \pm 50$ G, with an
inclination of $50^\circ \pm 13^\circ$.
