Conjecture on long screening lattices and Cartan-like roots

Let $\Lambda$ be a fractional lattice with short basis $\alpha_i^{short}$, and let $\Lambda^{long}$ be the integral sublattice with associated long basis $\alpha_i^{long}$, chosen so that the corresponding exponentials have conformal weight $1$. Long-screening lattice conjecture. For a general Lie algebra, $\Lambda^{long}$ should equal $\sqrt{p}\,\Lambda_{\mathfrak{g}}^\vee$; in the general Nichols-algebra case, the Cartan-like roots should generate the corresponding root lattice. The simply-laced case is stated to be proved, while the general assertions remain conjectural.