A note on the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E \otimes E$ over a finite field

Let $F$ be a finite field of $char F = p$ and size $|F| = q$. Let $E$ be the unitary infinity dimensional Grassmann algebra. In this short note, we describe the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E_{k^{*}}\otimes E$, where $E_{k^{*}}$ is the Grassmann algebra with a specific $\mathbb{Z}_{2}$-grading. In the end, we discuss about the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded GK-dimension of $E_{k^*}\otimes E$ in $m$ variables.