The equivariant Hodge index conjecture for arbitrary weights

Let Setup be fixed. For a weight $v \colon P \to \mathbb{R}_{>0}$, let $C_v>0$ be a constant, and let $\mathcal{E}$ be a $\mathbb{G}$-equivariant torsion-free sheaf. The equivariant Hodge index inequality concerns the equivariant first Chern class $c_1^{\mathbb{T}}(\mathcal{E})$ and the weight class $v(\alpha_{\mathbb{T}})$. Equivariant Hodge index conjecture. For every weight $v \colon P \to \mathbb{R}_{>0}$, there exists a constant $C_v>0$ such that

$$(c_1^{\mathbb{T}}(\mathcal{E})^2 \cdot v(\alpha_{\mathbb{T}})) \cdot (v(\alpha_{\mathbb{T}})) \le C_v(c_1^{\mathbb{T}}(\mathcal{E}) \cdot v(\alpha_{\mathbb{T}}))^2.$$

If equality holds, then $\widetilde{\Psi}(c_1^{\mathbb{G}}(\mathcal{E}))=0 \in H^2_{\mathrm{dR},\mathbb{T}}(X)$. This is posed in the outlook as a generalization of the equivariant Hodge index theorem from the soliton weight to arbitrary weights; its resolution is not supplied here.