The q,t Catalan difference divisibility conjecture

Let $\mathcal{C}_n(q,t)$ and $\mathcal{C}'_n(q,t)$ be the two $q,t$-Catalan polynomials introduced in the source. Divisibility conjecture. We have

$$\mathcal{C}_n(q,t)-\mathcal{C}'_n(q,t)=(qt-1)f_n(q,t)$$

for some $f_n(q,t)\in\mathbb{Z}_{\geq 0}[q,t]$. The source states that this relation was verified for all $n\leq 10$; its general validity remains open there.