Important progress on this question has been made by Yuki Maehara, who shows in [Orientals as free $\omega$-categories](https://arxiv.org/abs/2102.09736) 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.