Let $\mathcal{M}$ be a semifinite von Neumann algebra and $\tau$ be a semifinite faithful normal trace on it. Let $T,P_1,P_2\in \mathcal{M}$, where $P_1,P_2$ are projections with $P_1\perp P_2$. Then, obviously, we have $\tau(E^{|T|}(1,\infty)) \ge \tau(E^{|P_1TP_1|}(1,\infty)) $ and $\tau(E^{|T|}(1,\infty)) \ge \tau(E^{|P_2TP_2|}(1,\infty)) $.
My question: Whether $$\tau(E^{|T|}(1,\infty)) \ge \tau(E^{|P_1TP_1|}(1,\infty)) + \tau(E^{|P_2TP_2|}(1,\infty))$$ or not?