Tensor product of dualizing complexes over a field

Let $k$ be a field, and let $X,Y$ be two locally noetherian $k$-schemes (respectively $k$-formal schemes) with dualizing complexes $R_X$ and $R_Y$ respectively. We show that $R_X \boxtimes_{k} R_Y$ (respectively its derived completion) is a dualizing complex over $X\times_{k} Y$ if and only if $X\times_{k} Y$ is locally noetherian of finite Krull dimension.