I recently came upon the following situation (think of $\mathbb{R}^3$ to simplify): let $S$ be a compact smooth surface with $K>0$ everywhere and define

$$Q=\frac{\sup_{p}\lambda_{1}(p)}{\inf_{p}\lambda_{2}(p)},$$ where $\sup_{p}\lambda_{1}(p)$ is supremum of the principal curvatures and $\inf_{p}\lambda_{2}(p)$ is the infimum. Let $C$ be a planar curve contained in $S$ and define $$q=\frac{\sup_{p}\kappa(p)}{\inf_{p}\kappa(p)},$$ where $\kappa(p)$ is the curvature of $C$ at $p$. Is it true that $$q\lesssim Q,$$ where $\lesssim$ means that the implicit constant is "as universal as possible" (it could depend on the diameter of $S$, on the regularity, etc, but not on any kind of curvature)? Even learning that this is false would be interesting. If $S=\mathbb{S}^{2}$, for instance, $q=Q=1$.