Let $(R,m,k)$ be a complete, reduced, local ring of dimension $1$. Let $\omega=\omega_R$ be the canonical module of $R$. Let $H_n(\omega) =\ell(m^n\omega/m^{n+1}\omega)$. It is known that:

1. $R$ has minimal multiplicity ($e(R) =\mu(m)$) iff $H_0=e-1$ and $H_i=e, i\geq 1$.

2. $R$ has almost minimal multiplicity ($e(R) =\mu(m)+1$) then $H_2=e$.

Can we prove similar statements? For instance, can we bound the first index $n$ such that $H_n(\omega)=e$ by $e-\mu(m)+1$?