This post involves several different concepts.
1: inverse design uses binary permittivity/refractive index. However it does not need users to import binary material.
2:importnk2 imports refractive index varying with space. As for what part is used will depend on FDTD region.
3: Refractive index monitor:
in the preview before simulation, it is not recalculated at the material interace. It simply uses staircase depending on which material takes more than 50% of a Yee cell.
After simulation, it will show what exactly is used i simulation, from the "Mesh refinement" method: https://optics.ansys.com/hc/en-us/articles/360034382614-Selecting-the-best-mesh-refinement-option-in-the-FDTD-simulation-object
Therefore, it is normal to have a gap in index preview. If you use finer mesh it will reduce the gap. No worry about the gap as this is a preview. Please note that for this discrete method, everyting must be discretized, and the geometry/material is no exception. of couse the finer the mesh and the better the accuracy.