A (1,1)-topos (i.e. an ordinary Grothendieck topos) is called localic if the following two equivalent conditions hold:

• It is the category of sheaves on a (0,1)-site with finite limits$^*$ (i.e. a meet-semilattice with a Grothendieck topology).
• It is a left exact localization of the category of presheaves on a (0,1)-category with finite limits (= a meet-semilattice).

The category of localic topoi is reflective in the category of all topoi, and equivalent to the category of locales.

For $0\le m\le n \le \infty$, Higher topos theory 6.4.5.7-8 shows that an $(n,1)$-topos is $m$-localic iff the analogue of the first condition holds: it is the category of $(n-1)$-sheaves on an $(m,1)$-site with finite limits. In 6.4.5.9 it is remarked that this is equivalent to it being a topological localization of the category of $\infty$-presheaves on an $(m,1)$-category with finite limits. The category of $m$-localic $(n,1)$-topoi is reflective in all $(n,1)$-topoi, and equivalent to the category of $(m,1)$-topoi.

When $n<\infty$, every localization of an $(n,1)$-topos is topological, so the two conditions are still equivalent, but this fails when $n=\infty$. So what can be said about the larger class of $(\infty,1)$-topoi that are merely a left exact localization of $\infty$-presheaves on some $(m,1)$-category with finite limits? I believe this is equivalently the class of cotopological localizations of $m$-localic $(\infty,1)$-topoi, or the class of $(\infty,1)$-topoi that have a generating set consisting of $(m-1)$-truncated objects. Does it have a name? Has it been studied?

$^*$ Edit: In the original version of this question I omitted the assumption of finite limits. This is irrelevant when $n<\infty$ but makes a difference when $n=\infty$; see the comments. I do want to ask about the case where finite limits are assumed, so I edited the question.