I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.

" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, then: $$\| \sup_{n\geq 1} A_n |f| \|_p \leq q \| f \|_p$$ "

On page 193, in the notes he mentions that the proof of this theorem for positive power bounded $T$ is still open.

My question, is this problem as of yet still open, and if this indeed the case, what are the attempts at reconciling this open question? any added literature over the years since the publication of Krengel's book?

P.S $$A_n f = A_n(T)f = n^{-1} \sum_{j=0}^{n-1} T^j f $$