The special-horn generation conjecture for special-horn trivial cofibrations

Let $\operatorname{SpHorn}$ be the set of special horn inclusions. A map of simplicial sets is bijective on $0$-simplices when it induces a bijection on its sets of $0$-simplices. Special-horn generation conjecture. The bijective-on-$0$-simplices trivial cofibrations in the special horn model structure are precisely

$${}^{\boxslash}(\operatorname{SpHorn}^{\boxslash}).$$

Equivalently, the special horn inclusions generate this class of trivial cofibrations. This is a parallel generation statement for the special horn model structure, following the conjecture about the Joyal model structure; the supplied text does not establish whether it is known or resolved.