I am confused. Let E,F,X be spectra with $E_*(L_FX)=0$. Do we have $E_*(L_FL_EX)=0$ where $L_E,L_F$ are Bousfield localizations?

@FrankScience Not in general. $L_F$ might not respect $E$-equivalences

This seems to be used in the proof of the fracture theorem in the talk, Prop 2.2.

I've only seen that theorem used under the assumption that $L_FL_E=0$, so I'm not overly worried but let's see if I can cook up either a proof of the stronger claim or a counterexample :)

I wonder whether the 1-truncatedness of the map $\operatorname{BGL}_1(\mathbb S)\to\prod_p\operatorname{BGL}_1(\mathbb S_p^\wedge)$ follows from these fracture squares.

It seems related to the functoriality of the Sullivan's arithmetic square applied to the subcategory of $X\simeq\mathbb S$.