Sharp growth-factor bound for accretive-dissipative matrices

Let $A=B+iC\in\mathcal A_n$, where $B$ and $C$ are the real and imaginary parts of an accretive-dissipative matrix, and suppose that

$$\max\{\kappa(B),\kappa(C)\}\le\omega.$$

Here $\kappa$ denotes the condition number and $\rho_n(A)$ the growth factor. Accretive-dissipative growth-factor conjecture. One should have

$$\rho_n(A)\le\frac{2(1+\omega^2)}{(1+\omega)^2},$$

and the constant $\frac{2(1+\omega^2)}{(1+\omega)^2}$ should be sharp. The preceding theorem proves the same lower bound but only an upper bound larger by a factor of $\sqrt2$; numerical experiments suggest that this factor is an artifact of the reduction rather than a genuine obstruction.