YES

TRS:

  active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
  mark(f(X1,X2)) -> active(f(mark(X1),X2))
  mark(g(X)) -> active(g(mark(X)))
  f(mark(X1),X2) -> f(X1,X2)
  f(X1,mark(X2)) -> f(X1,X2)
  f(active(X1),X2) -> f(X1,X2)
  f(X1,active(X2)) -> f(X1,X2)
  g(mark(X)) -> g(X)
  g(active(X)) -> g(X)

linear polynomial interpretations on N:

  active_A(x1) = x1
  active#_A(x1) = x1 + 2
  f_A(x1,x2) = x1 + 1
  f#_A(x1,x2) = 0
  g_A(x1) = x1 + 2
  g#_A(x1) = x1 + 1
  mark_A(x1) = x1
  mark#_A(x1) = x1 + 3

precedence:

  mark > active > f > g