Numerical characterization conjecture for polarized manifolds via sectional geometric and Δ-genera

Let $(X,\mathcal{L})$ be a polarized manifold of dimension $n\geq 3$. For each integer $i$ with $2\leq i\leq n-1$, let $K_X$ denote the canonical divisor, let $g_i(X,\mathcal{L})$ denote the $i$th sectional geometric genus, let $\Delta_i(X,\mathcal{L})$ denote the $i$th $\Delta$-genus, and let $\mathcal{L}^n$ be the top self-intersection number of $\mathcal{L}$. Numerical characterization conjecture. The following conditions are equivalent:

$$K_X=-(n-i)\mathcal{L}.$$ $$\Delta_i(X,\mathcal{L})=1\quad\text{and}\quad 2g_1(X,\mathcal{L})-2=(i-1)\mathcal{L}^n.$$ $$\Delta_i(X,\mathcal{L})>0\quad\text{and}\quad 2g_1(X,\mathcal{L})-2=(i-1)\mathcal{L}^n.$$ $$g_i(X,\mathcal{L})=1\quad\text{and}\quad 2g_1(X,\mathcal{L})-2=(i-1)\mathcal{L}^n.$$ $$g_i(X,\mathcal{L})>0\quad\text{and}\quad 2g_1(X,\mathcal{L})-2=(i-1)\mathcal{L}^n.$$

The conjecture seeks a numerical characterization of polarized manifolds satisfying $K_X=-(n-i)\mathcal{L}$, extending Fujita's characterization of Del Pezzo manifolds. The source presents this as the main theme of the paper; its resolution status is not specified here.