Nonexistence of iterated countable meets and joins in the nowhere centered quotient

Let $\beta=\bigwedge_{n_0<\omega}\bigvee_{n_1<\omega}\cdots\omega^{k-1}\times A_{(n_0,n_1,\ldots,n_{k-1})}$ be the indicated iterated operation on a family of sets $\{A_{\overline{x}}\mid\overline{x}\in\omega^k\}\subseteq[\omega]^\omega$, where $2\leq k<\omega$. Iterated-operation conjecture. For every integer $k$ with $2\leq k<\omega$, there exists such a family for which $\beta$ does not exist in $\mathcal{P}(\omega^k)/\mathcal{NC}^k$. This would identify a limitation on iterating countable joins and meets in the nowhere centered quotient; the supplied text does not state whether the claim is known or open.