holds in the upper barrier sense: for every $\varepsilon>0$, there exists an open neighbourhood $U_\varepsilon \subset M$ of $q$ and a function $\varphi_\varepsilon \in C^2(U_\varepsilon)$ such that
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\begin{align*}
\varphi_\varepsilon(q) &= r_p(q),\\
\varphi_\varepsilon(x) &\geq r_p(x) \qquad \text{for every } x \in U_\varepsilon,\\
\Delta \varphi_\varepsilon(q) &\leq (n-1)\operatorname{ct}_k(r_p(q))+\varepsilon.
\end{align*}