Pseudoeffective threshold characterization of projective spaces and hyperquadrics

Let $X$ be a smooth projective variety and $A$ an ample divisor on $X$. Let

$$\Delta(X,A)=\sup\{\varepsilon\in\mathbb{R}\mid \xi-\varepsilon\pi^*A\text{ is effective}\},$$

where $\xi=c_1(\mathcal{O}_{\mathbb{P}(T_X)}(1))$ is the tautological class on $\mathbb{P}(T_X)$ and $\pi:\mathbb{P}(T_X)\to X$ is the natural projection. Pseudoeffective threshold characterization. If $\Delta(X,A)\geq 1$, then

$$(X,A)\simeq(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(1))\quad\text{or}\quad(Q^n,\mathcal{O}_{Q^n}(1)),$$

where $(Q^1,\mathcal{O}_{Q^1}(1))=(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(2))$. The conjecture would characterize projective spaces and hyperquadrics by the pseudoeffective threshold of their tangent bundles. A cited theorem implies the corresponding conclusion when a section exists with equal twisting, but the threshold condition allows the supremum to be approached without such a section, and this remains the conjectural case.