TITLE: Counter-example to the existence of left Bousfield localization of combinatorial model category
QUESTION [15 upvotes]: Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ?
It is well known to exists when $C$ is left proper, and it seems that it also always exists as a left semi-model structure, but I don't known if there is any concrete example where it is known to not be a Quillen model structure.
PS: I technically already asked this question a year ago but it was mixed with other related questions and this part was not answered, so I thought it was best to ask it again as a separate question.

REPLY [5 votes]: While preparing the paper Left Bousfield localization without left properness, I learned another example, due to Voevodsky. It's example 3.48 in his paper Simplicial radditive functors. This is an example of a model category that's not left proper, and a left Bousfield localization that does not exist as a model category. It does, however, exist as a semi-model category, as I prove in my paper.