Another determinantal inequality involving partial traces

Let $A$ be a positive semidefinite $m\times m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds $$(\mathrm{tr} A)^{mn} - \det(\mathrm{tr}_2 A)^n \ge \bigl| \det A - \det(\mathrm{tr}_1 A)^m \bigr|,$$ where $\mathrm{tr}_1$ and $\mathrm{tr}_2$ stand for the first and second partial trace, respectively. This result improves a recent result of Lin [14].