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).