Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category. For a diagram $F : \mathbb{C} \to \mathcal{M}$ and a weight $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, let $B_{\bullet} (G, \mathbb{C}, F)$ denote the two-sided simplicial bar construction, given in level $n$ by the formula below: $$B_n (G, \mathbb{C}, F) = \coprod_{(c_0, \ldots, c_n)} (G c_n \times \mathbb{C} (c_{n-1}, c_n) \times \cdots \times \mathbb{C}(c_0, c_1)) \odot F c_0$$ If I'm not mistaken, $B_\bullet (G, \mathbb{C}, F)$ is Reedy-cofibrant when $F c$ is cofibrant for all $c$ in $\mathbb{C}$.
Question. Is the statement correct? Is there a proof in the literature?
The special case where $G$ is constant with value $\Delta^0$ is proved as Lemma 5.2.1 in [Riehl, Categorical homotopy theory] and essentially boils down to the observation that the $n$-th latching object for $B_{\bullet} (G, \mathbb{C}, F)$ is the part of the coproduct indexed over the degenerate $n$-simplices of the nerve $N(\mathbb{C})$. As far as I can tell the same proof also works for the general case, except for having to invoke axiom SM7 to ensure that the summands of $B_n (G, \mathbb{C}, F)$ are cofibrant in $\mathcal{M}$.