Are the reduced horn maps R1(\tilde{J}) a generating set of acyclic cofibrations in 1-reduced simplicial sets? (Conjectured no)

Show that the set R1(\tilde{J}) = { R1(j_{n,k}) : R1(Λ[n,k]) → R1(Δ[n]) for all integers n ≥ 3 and 0 ≤ k ≤ n }, obtained by applying the 1-reduction functor R1 to the horn inclusions j_{n,k}, is not a generating set of acyclic cofibrations for the model category of 1-reduced simplicial sets SSets_1 (whose weak equivalences are weak homotopy equivalences and whose cofibrations are monomorphisms).

Background

In transferring the standard model structure from pointed simplicial sets to the category of 1-reduced simplicial sets SSets_1, the authors identify a generating set of cofibrations R1(I). For acyclic cofibrations, directly applying 1-reduction to all horn inclusions R1(J) fails because the n=2 horns do not yield weak equivalences after reduction.

Restricting to higher horns, they define \tilde{J} = { j_{n,k} : Λ[n,k] → Δ[n] | n ≥ 3, 0 ≤ k ≤ n } and show that R1(\tilde{J}) consists of acyclic cofibrations in SSets_1, detects fibrant objects, and detects fibrations whose targets are Kan complexes. However, it remains unresolved whether R1(\tilde{J}) actually generates all acyclic cofibrations in SSets_1. The authors conjecture that it does not.