The q,t-symmetry conjecture for two-part partitions

Let $a$, $n$, and $s$ be integers, and let

$$\lambda=((a+1)^s,a^{n-s})$$

be a partition consisting of $s$ copies of $a+1$ and $n-s$ copies of $a$. Let $C_{\lambda}(q,t)$ be the polynomial associated to $\lambda$ in the paper.

q,t-symmetry conjecture. The polynomial $C_{\lambda}(q,t)$ is $q,t$-symmetric.

The symmetry is known when the length of $\lambda$ is $2$, while the case of length $3$ was expected to be addressed in future work. The conjecture is motivated by computer experiments; the symmetry does not hold for general partitions with $n\geq 4$.