Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in *A folk model structure on omega-cat* left proper ?

All its objects are fibrant, so it is right proper, but left properness is not clear at all. I tend to think it is not, but the pushout of a cofibration in $\infty$-Cat are still a relatively well-behaved operation so it might very well be. In any case, I don't know explicit examples of pushouts showing it is not left proper.

I would also be interested in answers to this question for the folk model structure for 2 or 3-categories (either strict or semi-strict) as this would very probably shed some light on the question.

*Motivation:* I've stumbled on this for several reasons, but one of them is that the folk model structure has many interesting left Bousfield localizations. For example for each $n$ it can be Bousfield localized so that the fibrant objects of the localization are the $(\infty,n)$-category in the sense that all arrows of dimension $>n$ are weakly invertible (contrary to Ara–Métayer's "generalized Brown-Golanski" model structure in *The Brown-Golasinski model structure on strict $\infty$-groupoids revisited*, where the arrows of dimension $>n$ are strictly invertible; whether these two things are Quillen equivalent is as far as I know an open problem).

But in order for the localization to be an actual Quillen model category (instead of a semi-model category) one need left properness.

I'm very curious to know if this localization is an example of a Bousfield localisation of non proper model structure (because those have surprising properties and I don't know really natural examples of this so far), or if it is actually a Quillen model structure (which would also be nice).

suspectthe canonical model structure on $Cat_\omega$ is projectively-transferred along this adjunction from the Verity model structure for complicial sets. Verity's model structure is left proper, so this should imply that the canonical model structure is also left proper, right? $\endgroup$ – Tim Campion♦ Apr 10 at 14:53