Is there an argument in the litterature that show that every locally presentable $\infty$-category is equivalent to the localization of proper combinatorial Quillen model category ?

Of course if one removes proper this is well known, so equivalently I'm asking for a proof that every combinatorial model category equivalent to a proper one.

If one justs asks for right proper this is covered by Nikolaus construction (see corollary 2.21 here). For left proper this can be achieved using the "dual" version of this construction by Ching and Riehl (at least in the simplicial case).

Another way to get a left proper model is through the general construction of a model category from a locally presentable $\infty$-category:

If $\mathcal{E}$ is a locally presentable $\infty$-category, then it is an accessible localization of a presheaf category $Prsh(D)$. One then takes a model of $D$ as a (cofibrant) simplicial category and build a model for $C$ as a left Bousfield localization of the injective model category of simplicial presheaves on $D$. As a localization of a proper model category, it will be left proper.

Unfortunately, these are in general not right proper (this is related to whether $\mathcal{E}$ is cartesian closed).

I think it is actually possible to build a proper model as follows: The previous construction gives a model category which is left proper and where every cofibration is a monomorphism, it seems that in this case, a careful analysis of the construction in Nikolaus paper, show that the resulting model structure is actually left proper (and it is right proper as mentioned above). However this observation requires a significant amount of work.

So:

• Is there a published reference that would allow to justify the left properness of Nikolaus construction in this case ?

• Is there another (maybe simpler) way to get proper models (preferably in the literature)?

Is there an argument in the litterature that show that every locally presentable $\infty$-category is equivalent to the localization of proper combinatorial Quillen model category ?

Of course if one removes proper this is well known, so equivalently I'm asking for a proof that every combinatorial model category equivalent to a proper one.

If one justs asks for right proper this is covered by Nikolaus construction (see corollary 2.21 here). For left proper this can be achieved using the "dual" version of this construction by Ching and Riehl (at least in the simplicial case).

Another way to get a left proper model is through the general construction of a model category from a locally presentable $\infty$-category:

If $\mathcal{E}$ is a locally presentable $\infty$-category, then it is an accessible localization of a presheaf category $Prsh(D)$. One then takes a model of $D$ as a (cofibrant) simplicial category and build a model for $C$ as a left Bousfield localization of the injective model category of simplicial presheaves on $D$. As a localization of a proper model category, it will be left proper.

Unfortunately, these are in general not right proper (this is related to whether $\mathcal{E}$ is cartesian closed).

I think it is actually possible to build a proper model as follows: The previous construction gives a model category which is left proper and where every cofibration is a monomorphism, it seems that in this case, a careful analysis of the construction in Nikolaus paper, show that the resulting model structure is actually left proper (and it is right proper as mentioned above). However this observation requires a significant amount of work.

So:

• Is there a published reference that would allow to justify the left properness of Nikolaus construction in this case ?

• Is there another (maybe simpler) way to get proper models (preferably in the literature)?