I think that the  part $(a)$ of  proposition $2.4$ of [this paper ](https://arxiv.org/pdf/1506.08962.pdf)  shows that for $n$ sufficiently  large one  can construct  a  unitary  matrix $U=-I_{2}\oplus U'$,  which can  be  decomposed as the  product of three positive  matrices. For  such  $U$ we have $\parallel U-I\parallel=2$.

In fact  if $U=ABC$  for  positive  $A,B,C$ then $UC^{-1}=AB$  so $C^{-1}=|AB|$.