Non-generation conjecture for 1-reduced simplicial sets

Let $R_{1} \colon \mathcal{SS}\mathrm{ets} \rightarrow \mathcal{SS}\mathrm{ets}_{1}$ denote the $1$-reduction functor. For the set

$$R_{1}(\widetilde{J})=\{R_{1}(j_{n,k}) \colon R_{1}(\Lambda[n,k]) \rightarrow R_{1}(\Delta[n]) \mid n \geq 3 \textrm{ and } 0 \leq k \leq n\},$$

Non-generation conjecture. The set $R_{1}(\widetilde{J})$ is not a generating set of acyclic cofibrations for $\mathcal{SS}\mathrm{ets}_{1}$. The question is motivated by the fact that this set detects fibrations whose target is a Kan complex, while it is unknown whether it generates all acyclic cofibrations in the model structure on $1$-reduced simplicial sets.