Gorsky–Hogancamp–Simental–Rasmussen conjecture for positroid Catalan invariants

Let $f\in\Theta_{k,n}$ be repetition-free. Associate to $f$ the sequence $\mathbf{a}(f)$, the Coxeter link $\hat\beta(\mathbf{a}(f))$, the knot $\hat\beta_f$, the mixed Hodge polynomial $\mathcal{P}(\mathcal{X}^\circ_f;q,t)$, and the polynomials $C_{\mathbf{a}(f)}(q,t)$ and $\widetilde R_f(q)$.

Gorsky–Hogancamp–Simental–Rasmussen conjecture. The knots $\hat\beta(\mathbf{a}(f))$ and $\hat\beta_f$ are isotopic. Up to a monomial in $q$ and $t$,

$$\mathcal{P}(\mathcal{X}^\circ_f;q,t)=C_{\mathbf{a}(f)}(q,t).$$

Up to a monomial in $q$,

$$\widetilde R_f(q)=C_{\mathbf{a}(f)}(q,t=1/q).$$

The polynomials $\mathcal{P}(\mathcal{X}^\circ_f;q,t)$, $C_{\mathbf{a}(f)}(q,t)$, and $\widetilde R_f(q)$ have positive integer coefficients.

These claims relate positroid geometry, generalized $q,t$-Catalan numbers, and knot invariants from Khovanov–Rozansky homology. The source gives computational evidence and notes agreement at $q=t=1$, but does not state that the conjecture has been proved in general.