Starting from 2^n + 8^n + 9^n + 15^n = 3^n + 5^n + 12^n + 14^n for n = 1,2,3; apply GDM to get:
1 x 2^n + 2 x 8^n + 3 x 9^n + 4 x 15^n = 1 x 3^n + 2 x 5^n + 3 x 12^n + 4 x 14^n for n = 1,2; multiply both sides of this equation by 2 to get:
2(1 x 2^n + 2 x 8^n + 3 x 9^n + 4 x 15^n) = 2(1 x 3^n + 2 x 5^n + 3 x 12^n + 4 x 14^n) for n = 1,2; break open parentheses and multiply through to get:
2 x 2^n + 4 x 8^n + 6 x 9^n + 8 x 15^n = 2 x 3^n + 4 x 5^n + 6 x 12^n + 8 x 12^n for n = 1,2; now subtract the first equation from the last to get:
1 x 2^n + 3 x 8^n + 5 x 9^n + 7 x 15^n = 1 x 3^n + 3 x 5^n + 5 x 12^n + 7 x 14^n for n = 1,2
So if GDM works with a multigrade (which is dot multiplication by 1,2,3,4 in this case) then you can convert the coefficients by simple algebraic manipulation into odd numbers, or 1,4,7,10 or 4,3,2,1 or any consecutive series to dot multiply with as long as the multigrade is symmetric being dot multiplied by the same number of terms as in the multigrade which may produce a result that can work for n = 1 or n = 1,2 (but as my conjecture or proposition asserts, never works for n>2).