  We study some of the main properties (masses and open-flavor strong decay
widths) of $4P$ and $5P$ charmonia. While there are two candidates for the
$\chi_{\rm c0}(4P,5P)$ states, the $X(4500)$ and $X(4700)$, the properties of
the other members of the $\chi_{\rm c}(4P,5P)$ multiplets are still completely
unknown. With this in mind, we start to explore the charmonium interpretation
for these mesons. Our second goal is to investigate if the apparent mismatch
between the Quark Model (QM) predictions for $\chi_{\rm c0}(4P,5P)$ states and
the properties of the $X(4500)$ and $X(4700)$ mesons can be overcome by
introducing threshold corrections in the QM formalism. According to our
coupled-channel model results for the threshold mass shifts, the $\chi_{\rm
c0}(5P) \rightarrow X(4700)$ assignment is unacceptable, while the $\chi_{\rm
c0}(4P) \rightarrow X(4500)$ or $X(4700)$ assignments cannot be completely
ruled out.
