The l-Fano characterization conjecture for projective space

Let $X$ be an $n$-dimensional smooth $l$-Fano variety, meaning that $X$ is smooth and Fano and its Chern characters $\text{ch}_i(X)$ are positive for every $2\leqslant i\leqslant l$, where positivity means

$$\text{ch}_i(X)\cdot Z>0$$

for every effective $i$-cycle $Z$. The l-Fano characterization conjecture. If $l\geqslant\lceil\log_2(n+2)\rceil$, then

$$X\simeq\mathbb{P}^n.$$

The conjecture seeks to characterize projective space among smooth $l$-Fano varieties once the Chern-character positivity index is sufficiently large relative to the dimension. The source attributes it to earlier work, but provides no information about its resolution.