Modified positivity conjecture for $L$-classes of smooth complete intersections

Let $X$ be a smooth complete intersection of dimension $n$ and multidegree $(d_1,\ldots,d_r)$, with $r\geq 2$, and let $L^{2j}(X)$ denote its degree-$4j$ cohomology $L$-class. Assume, at least, that

$$\sum_{i=1}^r d_i\geq n+r.$$

Modified positivity conjecture for complete intersections. For every integer $j$ with $0\leq j\leq n/2$,

$$(-1)^jL^{2j}(X)>0.$$

This is a proposed sign pattern for the cohomology $L$-classes of smooth complete intersections under the stated degree condition. The source gives no evidence of a resolution.