Hawkes's maximal Dyck path formula for rational $q,t$-Catalan polynomials

Let $r$ and $s$ be relatively prime positive integers, and let $\mathbf{D}_{r/s}$ be the set of lattice paths from $(0,0)$ to $(s,r)$ using North and East steps that stay below the line joining these points. For $x\in\mathbf{D}_{r/s}$, let $area(x)$ and $dinv(x)$ be the rational Dyck path statistics, and define

$$\mathcal{C}_{r/s}=\sum_{x\in\mathbf{D}_{r/s}}q^{area(x)}t^{dinv(x)}.$$

For integers $a\leq b$, write $[a,b]_{q,t}=q^at^b+q^{a+1}t^{b-1}+\cdots+q^{b}t^a$, and write $(a,b)_{q,t}=[a+1,b-1]_{q,t}$. Let $\mathbf{T}_{r/s}\subseteq\mathbf{D}_{r/s}$ be the subset of maximal paths, meaning paths that pass as close as possible to the bounding diagonal. Partition it as $\mathbf{T}_{r/s}=\mathbf{T}_{r/s}^{+}\sqcup\mathbf{T}_{r/s}^{-}$, where $x\in\mathbf{T}_{r/s}^{+}$ if $area(x)\leq dinv(x)$ and $x\in\mathbf{T}_{r/s}^{-}$ otherwise.

Hawkes's conjecture. The rational $q,t$-Catalan polynomial satisfies

$$\mathcal{C}_{r/s}=\sum_{x\in\mathbf{T}_{r/s}^{+}}[area(x),dinv(x)]_{q,t}-\sum_{x\in\mathbf{T}_{r/s}^{-}}(dinv(x),area(x))_{q,t}.$$

The formula is symmetric in $q$ and $t$ by construction and gives a proposed combinatorial explanation for the famously open symmetry of rational $q,t$-Catalan polynomials. The paper provides theoretical and computational evidence and proves the formula in certain cases, but the general conjecture remains open.