A matrix $A\in M_{4}(\mathbb{C})$ is called a simple tensor if $A=B\otimes C$ for two $2\times 2$ matrices $B,C$.

Assume that $X$ is a Hausdorff topological space.Assume that $f:X\to M_{4}(\mathbb{C})$ is a continuous map such that $f(x)$ is a simple tensor, for every $x\in X$.

Are there continous maps $g,h:X\to M_{2}(\mathbb{C})$ with $f=g\otimes h$?