Let $V$ be a complex projective variety and $L$ a nef line bundle on $X$ (i.e., $L$ has non-negative intersection number with every cycle on $X$).

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