Lower bound of the parabolic Hilbert commutator

Answering a key point left open in recent work of Bongers, Guo, Li and Wick, we provide the following lower bound $$ \|b\|_{\text{BMO}_{\gamma}(\mathbb{R}^2)}\lesssim \|[b,H_{\gamma}]\|_{L^p(\mathbb{R}^2)\to L^p(\mathbb{R}^2)}, where $H_{\gamma}$ is parabolic Hilbert transform.