The framed ADKMV conjecture for the framed topological vertex

Let $\bm a=(a_1,a_2,a_3)\in\mathbb Z^3$, and let $\mu^1,\mu^2,\mu^3$ be three partitions. Let $W_{\mu^1,\mu^2,\mu^3}^{(\bm a)}(q)$ be the framed topological vertex, and define

$$A_{mn}^{ij}(q;\bm a)=q^{\frac{a_i m(m+1)-a_j n(n+1)}{2}}A_{mn}^{ij}(q).$$

Framed ADKMV conjecture. The framed topological vertex is given by

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

This generalizes the ADKMV conjecture from the unframed to the framed topological vertex. The supplied text reports that the one-legged case with $\mu^2=\mu^3=(0)$ and $a_2=a_3=0$, and the two-legged case with $\mu^3=(0)$ and $a_3=0$, have been proved; the general framed conjecture remains open on the supplied evidence.