The left-closedness conjecture for semidirect products of monoidal categories

Let $\mathcal{X}$ and $\mathcal{C}$ be left closed monoidal categories, meaning that tensoring on the left has a right adjoint. Let there be a strong action of $\mathcal{X}$ on $\mathcal{C}$. Left-closedness conjecture. The semidirect product $\mathcal{X} \ltimes \mathcal{C}$ is a left closed monoidal category. This conjecture is false; the paper disproves it and then produces examples of semidirect product monoidal categories that are left closed but not right closed.