YES

TRS:

  a__g(X) -> a__h(X)
  a__c() -> d()
  a__h(d()) -> a__g(c())
  mark(g(X)) -> a__g(X)
  mark(h(X)) -> a__h(X)
  mark(c()) -> a__c()
  mark(d()) -> d()
  a__g(X) -> g(X)
  a__h(X) -> h(X)
  a__c() -> c()

max/plus interpretations on N:

  a__g_A(x1) = max{1, 3 + x1}
  a__g#_A(x1) = max{1, 3 + x1}
  a__h_A(x1) = max{2, 1 + x1}
  a__h#_A(x1) = max{2, 1 + x1}
  a__c_A = 8
  a__c#_A = 8
  d_A = 7
  d#_A = 7
  c_A = 3
  c#_A = 3
  mark_A(x1) = max{9, 6 + x1}
  mark#_A(x1) = max{9, 6 + x1}
  g_A(x1) = max{0, x1}
  g#_A(x1) = max{0, x1}
  h_A(x1) = max{2, x1}
  h#_A(x1) = max{2, x1}

precedence:

  d > a__c > mark = g > a__g = a__h > c = h