The sharp Fischer-type determinantal inequality for accretive-dissipative matrices

Let $A\in \mathbb{M}_{n}(\bold C)$ be an accretive-dissipative matrix, partitioned as

$$A=\begin{pmatrix}A_{11}&A_{12}\A_{21}&A_{22}\end{pmatrix},$$

where $A_{11}$ is an $m\times m$ principal block and $A_{22}$ is the complementary block. Sharp Fischer-type determinantal inequality. One should have

$$|\det A|\leq 2^m|\det A_{11}|\,|\det A_{22}|.$$

The preceding argument proves the same inequality with the larger factor $2^{3m/2}$, so this conjecture asks whether the constant can be reduced to $2^m$.