Does External Tensor of Vector Bundles Preserve Zero Divisors

Question

Here $X$ is a CW complex. Suppose I have two vector bundles $V_1 \to X$ and $V_2 \to X$ such that $V_1 \otimes V_2 = 0$ in $K(X)$. Is it then true that $V_1 \boxtimes V_2 \to X \times X = 0$ in $K(X^2)$? My issue is that I am having a hard time thinking about $p_1^*(V_1) \otimes p_2^*(V_2)$ where $p_1$ and $p_2$ are the two projections $X \times X \to X$. Is there a nice description or way of thinking about it? Thanks.