The reduced LMOV conjecture for the framed unknot

Let $\tau\in\mathbb{Z}$ be a framing and let

$$Z_{\tau}(q,x)=\sum_{n\geq 0}\frac{(-1)^{n(\tau-1)}q^{\frac{n(n-1)}{2}\tau+\frac{n^2}{2}}}{(1-q)(1-q^2)\cdots(1-q^n)}x^n$$

be the reduced open string partition function of $(\mathbb{C}^3,D_{\tau})$. The reduced LMOV conjecture. There exist nonnegative integers $N_{m,k}(\tau)$, with only finitely many nonzero for each fixed $m\geq1$, such that

$$Z_{\tau}(q,x)=\prod_{m\geq1}\prod_{k\in\mathbb{Z}}\prod_{l\geq0}\left(1-q^{\frac{k}{2}+l}x^m\right)^{N_{m,k}(\tau)}.$$

This is a reduced, weak form of the LMOV integrality conjecture for the framed unknot; the supplied text does not state whether it has been proved or disproved.