$H\mathbb{Z}/4$ is not an $\mathbb{E}_2$-Thom Spectrum over the $2$-Complete Sphere Spectrum

For any prime number $p$, the classic Hopkins-Mahowald theorem asserts that $H\mathbb{Z}/p$ is an $\mathbb{E}_2$-Thom spectrum over the $p$-complete sphere spectrum. Kitchloo extended this result to show that $H\mathbb{Z}/p^k$ is an $\mathbb{E}_2$-Thom spectrum over the $p$-local sphere spectrum, ex