Does $\mathbf{Tos}$ generate $\mathbf{DLat}$ as a variety?

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Let $\mathbf{DLat}$ denote the variety of distributive lattices and let $\mathbf{Tos}$ denote the subclass of $\mathbf{DLat}$ consisting of the totally-ordered sets.

Question. Does $\mathbf{Tos}$ generate $\mathbf{DLat}$ as a variety?