Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg_LX = c_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $V$, considered as a prime cycle of dimension $\dim{X}$.

Question. For subvarieties $X$ and $Y$ of $V$ with $\deg_LX = \deg_LY = 0$, does it follow that all positive-dimensional components $Z$ of $X \cap Y$ necessarily have $\deg_LZ = 0$?