DT/PT/GW correspondence for Hilbert schemes in K-theory

Let $KDT(\mathbb P^1\times\mathbb C^2)$ denote the K-theoretic Donaldson–Thomas invariants of $\mathbb P^1\times\mathbb C^2$, and let $QK_{\mathrm T}(\operatorname{Hilb})$ denote the equivariant quantum K-theory of the Hilbert scheme of points. K-theoretic DT/PT/GW correspondence. K-theoretic DT invariants on $\mathbb P^1\times\mathbb C^2$ are in one-to-one correspondence with the quantum K-theory of the Hilbert scheme of points:

$$KDT(\mathbb P^1\times\mathbb C^2)\longleftrightarrow QK_{\mathrm T}(\operatorname{Hilb}).$$

The claim is presented as a proposed DT/PT/GW-type relationship for Hilbert schemes; the supplied text does not state whether it has been proved or disproved.