ct.category theory – tensor triangulated categories

Let $(mathcal{T},otimes)$ be a monoidal (not necessarily symmetric and possibly without unit object) triangulated category where $-otimes- $ is exact on both variables. Let $S$ be a set of objects in $mathcal{T}$ such that for any $s_1in S$ and $s_2in S$ we have that $s_1otimes s_{2}in S$.

I was wondering if the thick subcategory generated by $S$ is automatically monoidal?