If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into $\mathscr{L}_\infty$-spaces. To be more precise, if $W,Z$ are $\mathscr{L}_\infty$-spaces and $T\colon X\to W$, $S\colon Y\to Z$ are quotient maps then so is $T\otimes S$.
Now my question is: does a something similar hold for the spatial (injective) tensor product of operator spaces? The class replacing $\mathscr{L}_\infty$-spaces is presumably the class of nuclear operator spaces.