On the cardinality of interval Int(Pol$_k$) in partial $k$-valued logic

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$_k$. In this paper, we prove that if $k$ is divisible by the square of a prime number, then the family Int(Pol$_k$) contains an infinitely increasing (with respect to inclusion) chain of different closed classes. This result and the results obtained by the author earlier imply that the family Int(Pol$_k$) contains a finite number of closed classes if and only if $k$ is a prime number or a product of two different primes.