The X-basis conjecture for the Ding–Iohara Fock space

Let $m$ be a positive integer, let ${\boldsymbol \lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})$ be an $m$-tuple of partitions, and let $\mathcal{F}_{\mathbf{u}}$ and $\mathcal{F}_{\mathbf{u}}^*$ be the level-$m$ Fock space and its dual. The vectors $\left|X_{\boldsymbol \lambda}\right>$ and covectors $\left<X_{\boldsymbol \lambda}\right|$ are defined from the Fourier components $X_i^{(k)}$ of the operators $X^{(k)}(z)$ as in the preceding construction. X-basis conjecture. The family $\left(\left|X_{\boldsymbol \lambda}\right>\right)$ is a basis of $\mathcal{F}_{\mathbf{u}}$, and the family $\left(\left<X_{\boldsymbol \lambda}\right|\right)$ is a basis of $\mathcal{F}_{\mathbf{u}}^*$. This asserts a basis property for the vectors constructed from the Ding–Iohara algebra action; the source gives no resolution status or further evidence.