Property of almost independent images for ergodic transformations without partial rigidity

S. V. Tikhonov, in his paper of 2007 devoted to a new metric on the class of mixing transformations, faced the following natural question when studying the properties of such transformations: Does there exist a set $A$ with $\mu(A)=\frac12$ such that the inequality $|\mu(A\cap T^iA)-\mu(A)^2|<\varepsilon$ holds for all $i>0$? V. V. Ryzhikov (2009) obtained the following criterion: For an ergodic transformation $T$, a set $A$ of given measure such that $A$ and its images under $T$ are $\varepsilon$-independent exists if and only if $T$ does not possess the property of partial rigidity. The aim of the present study is to generalize this proposition to the case of multiple $\varepsilon$-independence of images.