YES

weighted path orer with linear polynomial on natural numbers:

  f_A(x1) = 3
  g_A(x1) = 2
  a_A = 1
  b_A = 0
  F_A(x1) = x1


precedence:

  a > F > b > f > g


status:
f: []
g: []
a: []
b: []
F: []

the condition F(a()) ->> F(g(a())) is unsatisfiable for all substitutions.