On the higher delta invariants of a Gorenstein local ring

Let $(R , \mathfrak {m} )$ be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by $\delta _R ^n(M)$ for each module $M$ and for each integer $n$. We propose a conjecture asking if $\delta _R ^n (R/\mathfrak {m} ^{\ell }) = 0$ for any positive integers $n$ and $\ell$. We prove that this is true provided the associated graded ring of $R$ has depth not less than $\operatorname {dim} R -1$. Furthermore we show that there are only finitely many possibilities for a pair of positive integers $(n, \ell )$ for which $\delta _R ^n (R/ \mathfrak {m} ^{\ell }) >0$.