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 $n \times n$ 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 one  can take a unitary  matrix $U'$  with  $Det \;U'=1$ such that $0$ lies in the interior of  the  convex hull of the eigenvalues of $U$. This convex hull is equal to $W(U)$, the numerical range of $U$.

In fact  if $U=ABC$  for  positive  $A,B,C$ then $UC^{-1}=AB$  so $C^{-1}=|AB|$ is the positive  factor of $U $ in its   polar  decomposition. Of  course such  $U$ has  $-1$ as an eigenvalue.