Positivity conjecture for cohomology $L$-classes of projective space

Let $n$ be a positive integer, let $\mathbb{P}^n$ be complex projective space, and let $L^{2j}(\mathbb{P}^n)\in H^{4j}(\mathbb{P}^n,\mathbb{Q})$ denote its degree-$4j$ cohomology $L$-class. Positivity conjecture for projective-space $L$-classes. For every integer $j$ with $0\leq j\leq n/2$,

$$L^{2j}(\mathbb{P}^n)>0.$$

This proposes positivity of all cohomology $L$-class components of projective space. The source gives no evidence of a resolution.