The Futaki upper-bound conjecture for toric Calabi–Yau threefolds

Let $\mathcal{X}$ be a toric Calabi–Yau $3$-fold with no factors of $\mathbb{C}$, let $\zeta_R$ be the Reeb symmetry, let $\zeta_p$ be the grading symmetry from GLSM-field degrees, and let $\eta_h$ be the test symmetry associated with a generator $x_h$. Futaki upper-bound conjecture.

$$F(\mathcal{X},\zeta_p,\eta_h)\leq \frac{8}{27}F(\mathcal{X},\zeta_R,\eta_h).$$

The inequality was observed for the 16 reflexive polygons and is proposed for all toric Calabi–Yau threefolds without a factor of $\mathbb{C}$; its general validity is open.