I was wondering in hollow-core fibres should the effective index of the fundamental mode be higher or less than the core's refractive index (in this case, n = 1 because the core is filled with air)?
For the simplest type of hollow-core fiber, you can find the answer in this paper by Marcatili and Schmeltzer, "Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission ". This paper originally appeared in Bell Systems Technical Journal in 1964 and has been made available for free download here: https://archive.org/details/bstj43-4-1783
According to Eq. (21) the refractive index is in fact smaller than unity.