[1] The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.
Some near-misses with high symmetry are also symmetrohedra with some truly regular polygon faces.
These polyhedra can be perturbed to become convex with faces that are arbitrarily close to regular polygons.
These cases use 4.4.4.4 vertex figures of the square tiling, 3.3.3.3.3.3 vertex figure of the triangular tiling, as well as 60 degree rhombi divided double equilateral triangle faces, or a 60 degree trapezoid as three equilateral triangles.
It is possible to take an infinite amount of distinct coplanar misses from sections of the cubic honeycomb (alternatively convex polycubes) or alternated cubic honeycomb, ignoring any obscured faces.