Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category.

Stable $\infty$-categories give an important example of a symmetric monoidal $(\infty,2)$-category where the tensor product of stable $\infty$-categories $C$ and $D$ is defined to be the stable infinity category $C \otimes D$ which is universal for functors out of $C \times D$ which are exact in each variable separately.

I gather that taking the homotopy category gives us a functor to the (ordinary, i.e. "(2,2)-category") of triangulated categories, triangulated functors, and transformations.

I am wondering if there is some symmetric monoidal structure on the category of triangulated categories so that "taking the homotopy" category becomes a (strong) symmetric monoidal functor? Can you recover the triangulated category $ho(C \otimes D)$ from the triangulated categories $Ho(C)$ and $Ho(D)$?