Let $K$ be an algebraically closed field. Let $X$ be an irreducible affine algebraic variety over $K$. Let $L/K$ be a field extension, where $L$ is also algebraically closed. Suppose the group of units of the coordinate ring $\mathscr{O}(X)$ is trivial (i.e. equals $K^\times$). Is it true that the group of units of $\mathscr{O}(X \times_K L)$ is also trivial (i.e. equals $L^\times$)?