Individual ergodic convergence conjecture for mean-bounded positive invertible operators

Let $\mathcal{M}$ be a von Neumann algebra, let $1<p<\infty$, and let $T:L_p(\mathcal{M})\to L_p(\mathcal{M})$ be a positive invertible operator with positive inverse. Suppose that

$$\sup_{n\in\mathbb Z}\left\|\frac{1}{2n+1}\sum_{k=-n}^n T^k\right\|<\infty.$$

Define the ergodic averages

$$A_n=\frac{1}{2n+1}\sum_{k=-n}^{n}T^k,\qquad n\in\mathbb N.$$

Mean-bounded ergodic convergence conjecture. The sequence $(A_nx)_{n\geq1}$ converges bilaterally almost uniformly to $Px$ for every $x\in L_p(\mathcal{M})$. If additionally $p\geq2$, $(A_nx)_{n\geq1}$ converges almost uniformly to $Px$.

This conjecture concerns individual ergodic convergence for mean-bounded positive invertible operators on noncommutative $L_p$-spaces. The corresponding result for classical $L_p$-spaces is known, while the noncommutative assertion is presented as still open.