Cartesian to geodetic conversion of 3D bounding box - How to calculate latitude and longitude from an axis aligned bounding box

I have a geometry with its vertices in cartesian coordinates. These cartesian coordinates are the ECEF(Earth centred earth fixed) coordinates. This geometry is actually present on an ellipsoidal model of the earth using wgs84 corrdinates.The cartesian coordinates were actually obtained by converting the set of latitudes and longitudes along which the geomtries lie but i no longer have access to them. What i have is an axis aligned bounding box with xmax, ymax, zmax and xmin,ymin,zmin obtained by parsing the cartesian coordinates (There is no obviously no cartesian point of the geometry at xmax,ymax,zmax or xmin,ymin,zmin. The bounding box is just a cuboid enclosing the geometry). Is there any way to get at least an approximate extents (min, max) of latitude and longitude using this bounding box?

The other idea is to reconvert all the cartesian points to latitude and longitude individually and then find the min, max from that but it is computationally too heavy.