Optimal torus manifold conjecture for multifans

Let $X$ range over real $2n$-dimensional torus manifolds corresponding to a multifan $\mathcal S_X$. Let $h_k(\mathcal S_X)=\sum_{i=k}^n(-1)^{i-k}\binom{i}{k}e_{n-i}(\mathcal S_X)$. Optimal torus manifold conjecture. Among the torus manifolds corresponding to $\mathcal S_X$, there exists an optimal torus manifold whose $\mathbb T^n$-equivariant Betti numbers are minimally non-negative and satisfy

$$b_{2k}(X)\geq h_k(\mathcal S_X),\qquad k=0,1,\ldots,n,$$

with minimal values obeying Poincaré duality $b_{2n-r}\leq b_r$, the Hard Lefschetz theorem, and consistency with all $\mathbb T^n$-equivariant characteristic classes encoded by $\mathcal S_X$. The proposal seeks a preferred torus manifold realizing the smallest admissible Betti data compatible with the multifan; the source gives no resolution.