Generalized complexity conjecture for generalized log Calabi–Yau pairs

Let $(X,B,\mathbf M.)$ be a generalized log Calabi–Yau pair. Write $|B|$ for the sum of the coefficients of $B$ and $|\mathbf M.|$ for the sum of the coefficients of the moduli part in its decomposition as a sum of nef Cartier divisors on a model where it descends.

Generalized complexity conjecture. The following statements hold:

$$\dim X+\rho(X)-|B|-|\mathbf M.|\geq 0.$$

2. If equality holds, then $(X,\lfloor B\rfloor)$ is toric.
3. If $B=0$, $\mathbf M.$ descends on $X$, and equality holds, then $X$ is a product of projective spaces.

This conjecture interpolates between the complexity characterization of toric varieties and the Kobayashi–Ochiai theorem. The paper presents it as an open conjecture; in particular, the generalized klt case motivates the expected product-of-projective-spaces conclusion, while the corresponding generalized log canonical statement can fail.