This is a problem that I have been stuck for a few months.

Let $X$ be a Hilbert space and $A:B:X\to X$ be two non-commuting semi-positive self-adjoint bounded linear operators. Is it true that $$\|(I+A+B)^{-1}A\|\le 1.$$ If it is, can you suggest any reference to me?

I was able to show $\|(I+A+B)^{-1}A\|\le \|A\|$ and $\|(I+A)^{-1}A\|\le 1$. However, I don't know how to show the desired result without assuming $\|A\|\le 1$ or $AB=BA$.

If it helps, I can show $A=C^*D_1^*D_1C$ and $B=C^*D_2^*D_2C$ for some linear bounded $C,D_1,D_2$.