The ADKMV conjecture for the topological vertex

Let $\mu^1,\mu^2,\mu^3$ be partitions, and let $W_{\mu^1,\mu^2,\mu^3}(q)$ denote the topological vertex. For $i,j\in\{1,2,3\}$ and $m,n\geq 0$, let $A_{mn}^{ij}(q)$ be the explicit coefficients specified by the combinatorial formulas in the source. In the three-component fermionic Fock space $\mathcal F^{(0)}\otimes\mathcal F^{(0)}\otimes\mathcal F^{(0)}$, write $|\mu^1,\mu^2,\mu^3\rangle=|\mu^1\rangle\otimes|\mu^2\rangle\otimes|\mu^3\rangle$. ADKMV conjecture. The topological vertex is given by

$$W_{\mu^1,\mu^2,\mu^3}(q)=\langle\mu^1,\mu^2,\mu^3|\exp\Big(\sum_{i,j=1,2,3}\sum_{m,n\geq0}A_{mn}^{ij}(q)\psi_{-m-\frac12}^{i}\psi_{-n-\frac12}^{j*}\Big)|0\rangle\otimes|0\rangle\otimes|0\rangle.$$

This conjecture identifies the topological vertex with a Bogoliubov transform and is motivated by the conjectural multi-component KP integrability of its generating function. The supplied text does not establish the full conjecture; only framed one- and two-legged cases are reported as proved elsewhere.