On closed classes in partial $k$-valued logic that contain all polynomials

Let $Pol_k$ be the set of all functions of $k$-valued logic representable by a polynomial modulo $k$, and let $Int(Pol_k)$ be the family of all closed classes (with respect to superposition) in the partial $k$-valued logic containing $Pol_k$ and consisting only of functions extendable to some function from $Pol_k$. Previously the author showed that if $k$ is the product of two different primes, then the family $Int(Pol_k)$ consists of 7 closed classes. In this paper, it is proved that if $k$ has at least 3 different prime divisors, then the family $Int(Pol_k)$ contains an infinitely decreasing (with respect to inclusion) chain of different closed classes.