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Krzysztof Worytkiewicz kindly informs me that an old question to Francois Metayer has now been answered: the folk model structure on strict $\omega$-categories does restrict to the Brown-Golasinski model structure on strict $\omega$-groupoids. (The latter is indeed the transferred model structure along the forgetful functor to the former).
This is now written up in
Ara, Métayer, The Brown-Golasinki model structure on oo-groupoids revisited (pdf)
As my connection allows, I will insert this into the nLab entry now…
So now the remaining question is: does the $\omega$-nerve constitute a full and faithful $\infty$-functor from strict $\omega$-groupoids into all $\infty$-groupoids?
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