"Any two distinct vertices"? So, anywhere at all? As a function of... what kind of distance? We need axes defined here, because the shape is not perfectly uniform (there is distortion from the original geometric shape).
Or did you mean any pair of vertices joined by an edge? In that case, we still must specify location somehow, because notice the original shape's vertices are always within five triangles, while the others (along edges and in faces) are within six.
Since the surface is mostly regular, the solution can be arrived at using another regular mathematical method: the Fourier transform. For instance, the knights-move on a square grid picks up a factor of pi this way.
In the limit as N vertices --> infty, the 12 original vertices are diluted to ~zero, so that you can instead analyze the resistance of "almost all" edges as that of an infinite (flat) triangle grid. (A sphere of infinite size has zero curvature, as any Flat Earther knows. :-) ) So, at this point, it doesn't matter what the original shape was.
For edges coincidentally very close to an original vertex (a finite number of edges away from the vertex), the five-triangle point looks like a defect, and will have a perturbative effect on the region around it (for which the Fourier method would probably be best).
As it happens, it looks like equivalent methods are needed anyway;
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Tim