I have the following situation: 

 1. Three combinatorial model categories $\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$ such that all objects are fibrant (so no need to bother with the fibrant replacements and also they are right proper).
 2. Two Quillen adjunctions $L_1\dashv R_1$ and $L_2\dashv R_2$ with $\mathcal{K}_1 \stackrel{L_1}\longleftarrow \mathcal{K}_2 \stackrel{L_2}\longrightarrow \mathcal{K}_3$
 3. $L_2\dashv R_2$ is a Quillen equivalence. Moreover $R_2$ is also a (categorical) left adjoint.
 4. There is a functor $F:\mathcal{K}_1\to \mathcal{K}_3$ which is neither a left nor a right adjoint; the functor $F.(-)^{cof}$ induces an equivalence of categories from $\mathrm{Ho}(\mathcal{K}_1)$ to $\mathrm{Ho}(\mathcal{K}_3)$
 5. $F=L_2.R_1$.
 6. $R_1$ and $R_2$ reflect weak equivalences.
 7. $\mathcal{K}_3$ is left proper.
 8. The three model categories $\mathcal{K}_1$, $\mathcal{K}_2$ and $\mathcal{K}_3$ are simplicial and tractable.

> Does it suffice to conclude that the Quillen adjunction $L_1\dashv R_1$ is a Quillen equivalence ?

The answer is likely to be negative unless I am missing a stupid point. Indeed, the Quillen adjunctions $L_i\dashv R_i$ yield the derived adjunctions $\mathbf{L}L_i\dashv \mathbf{R}R_i$ between the homotopy categories. And the hypothesis (5) does not imply anything about the composite $\mathbf{L}L_2.\mathbf{R}R_1$, unless $R_1$ takes cofibrant objects to cofibrant objects which is plausible in my situation but extremely difficult to prove directly.

> What kind of sufficient conditions could lead to this result, given
> the fact that the functor $L_1$ is also extremely complicated to understand ?