Being a loop antenna I’d imagine it unadvisable to use any ferrous material at all, including the braid of the coax which is running straight through the whole antenna.
With large diameter conductors, no tuning instructions, and no ultra-critical dimensions in the fabrication, I'd imagine this is not a terribly high-Q antenna. Early in the video the design wavelength is listed as 190mm - the 1575 MHz L1 band of GPS.
What would be the primary problem (if any) caused by a small amount of ferrous material inside this particular antenna design? A shift in resonance frequency, or unwanted nulls in certain directions of the radiation pattern, or just an overall loss of gain?