Lots of other point collisions too: {{{-1, 3/2, 13/2}, {1, -3/2, 13/2}, {0, 0, -1/4}}, {{0, 7/8, -(3479/1152)}, {48/7, 0, 49/1152}, {-(48/7), 7/16, 539/2304}}, {{-1, 7/8, -(40523/648)}, {160/7, 0, 49/5184}}, {{-2, 7/8, 2989/1296}, {32/21, 0, -(49/192)}}, {{1/63 (-120 - 16 Sqrt[3]), 7/8, 0}, {4/7 (-1 + 3 Sqrt[3]), 0, 49/288 (5 + 3 Sqrt[3])}} }; Next steps would be a simpler counterexample or family of polynomials where the conjecture breaks.