Lower bound of the parabolic Hilbert commutator

Answering a key point left open in a 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 the parabolic Hilbert transform.