Let $(R,m,k)$ be a complete, reduced, local ring of dimension $1$. Let $S_n(R) =\Omega_n(mod(R))$ (the category of $n$-syzygy modules). In this situation $S_2(R)$ is simply the category of reflexive modules. It is known that if $R$ is Arf, then $S_2(R)$ is of finite type (fintely many indecomposable objects upto isomorphism). In fact, the indecomposable are precisely isoclasses of the "infinitely near points".

What can we prove about $S_n(R)$ for higher $n$?