Consider the strict 2-category of monoidal categories with (lax) monoidal functors as 1-morphisms. There is a general notion of adjunctions and (adjoint) equivalences in (strict) 2-categories. Call these monoidal adjunctions resp. monoidal equivalences in our special case.
Let $L\vdash R$ be a monoidal equivalence. Is it true that $L$ and $R$ are strong monoidal? Note that there is an oplax structure on $F$ given by $$L(X\otimes Y)\longrightarrow L(RLX\otimes RLY)\longrightarrow LR(LX\otimes LY)\longrightarrow LX\otimes LY$$ (in fact, this only requires $L$ to be monoidally left adjoint to $R$), and similarly for $R$.
But I am not able to prove that this is inverse to the lax structure morphism of $L$. Is it even true?