A counterexample for some problem for de Groot dual iterations

We prove that there is a topology τ that does not arise as a de Groot dual topology such that τ d = τ d d d ≠ τ d d ⊊ τ (i.e. the answer for Question 3.9 [M.M. Kovár, At most 4 topologies can arise from iterating the de Groot dual, Topology Appl. 130 (2003) 175–182] is negative).