Does $A\le B$ imply an inequality between their distribution functions?

Let $(\mathcal M,\tau)$ be a finite von Neumann algebra with a faithful normal tracial state $\tau$, and let $A,B$ be self-adjoint $\tau$-measurable operators affiliated with $\mathcal M$. Suppose that $A\le B$ and $ 1_E(A)1_F(B)=1_F(B)1_E(A) $ for all Borel sets $E,F\subseteq\mathbb R$. I would like to know whether, for every $x\in\mathbb R$, one necessarily has $ \tau\left(1_{(-\infty,x]}(B)\right) \le \tau\left(1_{(-\infty,x]}(A)\right). $

I am aware of a result in the paper (Corollary 2.9) which states that, under appropriate assumptions, if $ 0\le A\le B, $ then $ \tau(f(A))\le\tau(f(B)) $ for suitable continuous increasing functions $f$ with $f(0)=0$.

I also know that if $0\le A\le B$ and $x>0$, then the above inequality for spectral projections is true. However, in my case, $x$ can be any real number, and $A$ and $B$ need not be positive.

I would therefore be grateful if someone could clarify whether the above inequality holds under the stated assumptions. In particular, I would also appreciate a reference if this result is known in the literature. Thank you.