Does Fermat’s principle require equal optical path lengths for all rays or just minimization for each individual ray? How this applies in a GRIN lens?

My problem is about rays traveling through a GRIN medium, where the refractive index is maximum at the center and decreases quadratically outward. Intuitively, I would expect rays starting above or below the axis to refract upward (away from the axis), since that seems like the way to minimize their own optical path length (shorter distance, lower index region). But in fact, the rays bend inward toward the axis. In the MIT OCW 2009 optics lecture 11 (Prof. George Barbastathis), the justification is that since the rays originate from a point at infinity, they must all have equal optical path length in order to form an image at the focal point. But in my case, I’m not sure whether the rays actually meet again — so the “equal OPL” condition seems to disappear. So my confusion is: Does Fermat’s principle apply per ray (each one minimizes its own optical path), or across rays (requiring equal OPL)? In a GRIN medium without a guaranteed image point, why do the rays still bend inward instead of outward if the goal is minimization of OPL? Figure from MIT OpenCourseWare, 2009, Prof. George Barbastathis.