Complementation in tensor products

This question, however looks innocent, looks non-trivial to me. Suppose that $X$ and $Y$ are Banach spaces and let $\alpha$ be any reasonable cross-norm on $X\otimes Y$. Reasonable means that $\|x\otimes y \| = \|x\|\|y\|$ for $x\in X, y\in Y$ and $\|f\otimes g\|=\|f\|\|g\|$ for $f\in X^*$ and $g\in Y^*$.

Can we conclude that the operator $I_X \otimes P\colon X\otimes_\alpha Y\to X\otimes_\alpha Y$ is bounded if $P$ is a rank-one projection onto any unit vector $y_0\in Y$ given by $Py = g(y)y_0$ where $g\in Y^*$ is such that $g(y_0)=1$?

Tensor products usually make me nervous, but this seems straightforward. Fix a finite sum $\sum x_j\otimes y_j$, then \begin{align*} \| I_X\otimes P(\sum x_j\otimes y_j) \|_\alpha &= \|\sum g(y_j)x_j \otimes y_0\|_\alpha \\ &= \|\sum g(y_j)x_j\|_X \|y_0\|_Y \\ &=\sup_{f\in X^*, \|f\|\leq 1} \left|\sum f(x_j)g(y_j)\right| \|y_0\|\\ &= \sup |(f\otimes g)(\sum x_j\otimes y_j)| \|y_0\| \\ &\leq \|g\|_{X^*} \|y_0\| \| \sum x_j\otimes y_j\|_\alpha. \end{align*}