Let $\mathcal{M}$ be a point model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, Model categories and their localizations, Ch. 3).

Assume $F\to E\to B$ is a fibre sequence and all three objects are fibrant (in $\mathcal{M}$). If $F, B$ are fibrant in ${\rm L}_{\mathcal{C}}\mathcal{M}$ (i.e. they are $\mathcal{C}$-local, Hirschhorn, def. 3.1.4), do we have that $E$ is also $\mathcal{C}$-local? Or can we replace this fibre sequence (in $\mathcal{M}$) by $F\to E'\to B$ with $E'$ $\mathcal{C}$-local?

Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, Model categories and their localizations, Ch. 3).

Assume $F\to E\to B$ is a fibre sequence and all three objects are fibrant (in $\mathcal{M}$). If $F, B$ are fibrant in ${\rm L}_{\mathcal{C}}\mathcal{M}$ (i.e. they are $\mathcal{C}$-local, Hirschhorn, def. 3.1.4), do we have that $E$ is also $\mathcal{C}$-local? Or can we replace this fibre sequence (in $\mathcal{M}$) by $F\to E'\to B$ with $E'$ $\mathcal{C}$-local?