They will not be able to boost back to 5V because MAX38888 will require the supply to be above VSYS max (5V) to start charging. The device requires VSYS to be 12% above the boost target to start charging. So, 4.65V boost target will also require VSYS to be above max 5V.
Since the Vsys is up to 5V only, so to keep VSYS within the limition in datasheet (5V max), the VSYS boost target must be below 4.5V.