The nonexistence conjecture for polynomial enumeration of divisor-count classes

For $k$ with $0<k<n$, define $A_{2k}=\{n\in\mathbb{N}:\nu(n)=2k\}$, where $\nu(n)$ is the number of positive divisors of $n$. The divisor-class polynomial conjecture. No polynomial in $n$ generates the elements of $A_{2k}$ successively. The claim is presented as an extension of the asserted nonexistence of a polynomial generating all primes successively, but the source gives no resolution evidence.