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.