Conjecture on the logarithmic extension generated by highest-root exponentials

Let $\mathrm{W}$ be the kernel intersection of the short and long screenings inside $\mathcal{V}_{\Lambda^{long}}$, and let $\mathcal{W}^{\log}$ be the larger subspace obtained by imposing only the short-screening kernel condition. Logarithmic extension conjecture. The subspace $\mathcal{W}^{\log}\supset\mathrm{W}$ should be a vertex subalgebra carrying an action of $\mathfrak{g}$ from the long screenings, generated by pure exponentials $\exp(\phi_{-\alpha})$ for highest roots $\alpha$, which span the adjoint representation of $\mathfrak{g}$. This is part of the proposed logarithmic conformal-field-theory construction, and no proof or disproof is given.