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In July last year I made this deltahedron. It can be constructed from two icosidodecahedra joined at a triangular face and excavated, so that the resulting shape has 234 equilateral triangle faces.
It has 351 edges, 237 of which are concave.

If you conjoin n >= 2 icosidodecahedra at triangular faces, then excavate, the proportion of concave edges is (117n+3)/(171n+9), which means you can make the proportion of concave edges of a deltahedron as close as you like to 116/171=13/19≈68.4%.