Ergodic variant of the quadratic Rado conjecture

Let $(a,b,c)$ be a Rado triple, meaning that $a=c$, $b=c$, or $a+b=c$. A multiplicative action is a quadruple $(X,\mathcal{X},\mu,T_n)$ where $(X,\mathcal{X},\mu)$ is a probability space and the invertible measure-preserving transformations $T_n$ satisfy $T_1=\operatorname{id}$ and $T_m\circ T_n=T_{mn}$. Let $A\in\mathcal{X}$ satisfy

$$X=\bigcup_{j=1}^kT_j^{-1}A$$

for some $k\in\mathbb{N}$.

Ergodic variant of the quadratic Rado conjecture. There exist distinct $x,y,z\in\mathbb{N}$ such that

$$\mu\bigl(T_x^{-1}A\cap T_y^{-1}A\cap T_z^{-1}A\bigr)>0$$

and

$$a x^2+b y^2=c z^2.$$

This multiple-recurrence statement is presented as an ergodic-theoretic reduction of the quadratic Rado conjecture. The cited results establish related pair configurations and special coefficient cases, but the stated triple recurrence problem remains open.