Rank-factorization conjecture for the polynomials $P_\lambda(z;q,t)$

Let $\lambda$ be a partition, and let its rank be the size of its Durfee square. Write $P_\lambda(z;q,t)$ for the polynomial associated with $\lambda$. Computations indicate that these polynomials have factors of the form $z+q^it^j$.

Rank-factorization conjecture. For any partition $\lambda$ of rank $r$, there is a polynomial $Q_\lambda(z;q,t)\in\mathbb{Z}_{\geq 0}[z,q,t]$ such that

$$P_\lambda(z;q,t)=Q_\lambda(z;q,t)\prod_{i,j\geq 0\,;\,i+j<r}(z+q^it^j).$$

Equivalently, $P_\lambda(z;q,t)$ is divisible by each of the displayed linear factors.

The conjecture specifies the linear factors of $P_\lambda(z;q,t)$ in terms of the Durfee-square rank of $\lambda$. It was motivated by computations for small partitions and has been verified for all partitions of size at most $9$; the general statement remains open.