Ergodic Pythagorean recurrence conjecture

A multiplicative measure-preserving system is a probability space $(X,\mu)$ equipped with an action $T=(T_n)_{n\in\mathbb{N}}$ of $(\mathbb{N},\times)$ such that every $T_n$ preserves $\mu$. Let $A\subset X$, and suppose that for some $r\in\mathbb{N}$,

$$\mu\left(A\cup T_1^{-1}A\cup\cdots\cup T_r^{-1}A\right)=1.$$

Ergodic Pythagorean recurrence conjecture. There exists a solution $(x,y,z)\in\mathbb{N}^3$ of $x^2+y^2=z^2$ such that

$$\mu\left(T_x^{-1}A\cap T_y^{-1}A\cap T_z^{-1}A\right)>0.$$

This is presented as an equivalent ergodic form of the monochromatic Pythagorean-triples conjecture via Furstenberg's correspondence principle. Since the underlying partition-regularity conjecture remains open, this equivalent formulation is also open.