Doesn't CAT III b antennae type allow landing down to 0 AGL?.. So, probably, precision depends on the type of equipment installed!
Equipment type for ILS CAT II or CAT III doesn't matter for a discussion related to CAT I ILS. Below the minimums you are to conduct the remainder of the approach visually; so even if the equipment has the precision to allow for lower minimums, that's not what you're supposed to be looking at.