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Important progress on this question has been made by Yuki Maehara, who shows in Orientals as free $\omega$-categories that the complicial nerve of the $n$th oriental is a complicial anodyne extension of the $n$-simplex (providing a fibrant replacement thereof in Verity's model structure on complicial sets). In other words, at least in the complicial set model, the orientals -- which a priori have a strict universal property -- have the expected weak universal property too.