Zero cycles on products of elliptic curves over local fields with supersingular reduction

For a product $E_1\times E_2$ of two elliptic curves over a $p$-adic field with good supersingular reduction, we produce infinitely many rational equivalences in the Chow group ${\mathrm{C}\mathrm{H}}_0(X)$ of zero cycles via genus 2 covers of $E_1$ and $E_2$. We use this to obtain evidence for a conjecture of Colliot-Thélène about the structure of the Albanese kernel.