This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"

In Part (2) of Theorem 19.8.4 of that book it says

If $(\bf{\Delta},\mathcal{M})$ is a Reedy framed diagram category structure on the category of cosimplicial objects in $\mathcal{M}$ and $\bf X$ is a cosimplicial object in $\mathcal{M}$, then the total object $\text{Tot}\bf X$ of $\bf X$ is naturally weakly equivalent to the homotopy limit $\text{holim} \bf X$ of $\bf X$.

I notice that in Part (1) of the same theorem there is a dual statement on homotopy colimit of simplicial object but it require that $\bf X$ is Reedy cofibrant.

My question is: could we really get rid of the cofibrancy requirement if we want to show that the totalization of a cosimplicial object is weakly equivalent to the homotopy limit?