The equality of combinatorial and geometric refined BPS invariants

For the local $\mathbb{P}^2$, let $\widetilde{n}_d^{i,j}$ denote the combinatorial refined BPS invariants obtained from the Nekrasov partition function, and let $n_d^{i,j}$ denote the refined BPS invariants defined via the perverse filtration. The refined BPS correspondence conjecture. For all relevant indices $d,i,j$, one has

$$\widetilde{n}_d^{i,j}=n_d^{i,j}.$$

The equality would identify the easily computable Nekrasov invariants with the geometric invariants from the moduli-space perverse filtration. The text presents this as expected from string-theoretic considerations, and no proof or disproof is supplied.