YES

TRS:

  f(x,y) -> g1(x,x,y)
  f(x,y) -> g1(y,x,x)
  f(x,y) -> g2(x,y,y)
  f(x,y) -> g2(y,y,x)
  g1(x,x,y) -> h(x,y)
  g1(y,x,x) -> h(x,y)
  g2(x,y,y) -> h(x,y)
  g2(y,y,x) -> h(x,y)
  h(x,x) -> x

max/plus interpretations on N:

  f_A(x1,x2) = max{4, 1 + x1, 2 + x2}
  f#_A(x1,x2) = max{4, 6, 8}
  g1_A(x1,x2,x3) = max{0, 1 + x1, 4, 1 + x3}
  g1#_A(x1,x2,x3) = max{5, 3, 3, 3}
  g2_A(x1,x2,x3) = max{4, 3, 2 + x2, 0}
  g2#_A(x1,x2,x3) = max{7, 1, 1, 1}
  h_A(x1,x2) = max{4, 1, 1 + x2}
  h#_A(x1,x2) = max{0, 0, 2}

precedence:

  f > g1 = g2 > h