YES

TRS:

  p(f(f(x))) -> q(f(g(x)))
  p(g(g(x))) -> q(g(f(x)))
  q(f(f(x))) -> p(f(g(x)))
  q(g(g(x))) -> p(g(f(x)))

max/plus interpretations on N:

  p_A(x1) = max{22, 2}
  p#_A(x1) = max{2, -12 + x1}
  f_A(x1) = max{1, 2}
  f#_A(x1) = max{3, -12}
  q_A(x1) = max{22, x1}
  q#_A(x1) = max{3, -12 + x1}
  g_A(x1) = max{13, 8 + x1}
  g#_A(x1) = max{0, -1}

precedence:

  p > f > q > g