Multiple recurrence conjecture for computable commuting transformations

Let $(X,\mu)$ be a computable probability space. Let $T_1,\ldots,T_k$ be computable measure-preserving transformations that commute pairwise. Let $P$ be a $\Pi^0_1$ class with $\mu(P)>0$. A point $z\in P$ is Martin-Löf random if it avoids every effectively null $\Sigma^0_1$ test. Multiple recurrence conjecture. If $z\in P$ is Martin-Löf random, then there exists $n$ such that

$$z\in\bigcap_{i\leq k}T_i^{-n}(P).$$

This is a proposed extension of multiple recurrence to computable probability spaces and computable pairwise commuting measure-preserving transformations. The supplied text calls the result “putative” and gives no evidence of a resolution.