Largest disk inside a spherical domain

Question

It is known (Pestov-Ionin theorem) that if $k_{max}$ is the maximum curvature of a smooth planar loop $\gamma$, then there is a disk of radius $1/k_{max}$ inside $\gamma$. I wonder is there any analogue of this result for loops on the sphere. Indeed, it cannot hold in exactly the same form as on the sphere one can have $k_{max}<0$...