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I am re-reading Simpson/Teleman's "de Rham theorem for oo-stacks" and realize what I missed on first reading:
they have essentually the statement that I produced recently, that oo-stacks on a site with "geometrically contractible" objects form a localy contractible (infinity,1)-topos in that the terminal geometric morphism is essential.
They say this (slightly implicitly) at the level of homotopy cateories on page 5.
I briefly recorded this in a somewhat stubby way now at locally contracible (oo,1)-topos - Examples
Added this remark also to homtopy groups in an (oo,1)-topos - Geometric version -- By local contractions -- References
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