On generalized Narita ideals

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$. An $\mathfrak{m}$-primary ideal $I$ is said to be a generalized Narita ideal if $e_i^I(A) = 0$ for $2 \leq i \leq d$. If $I$ is a generalized Narita ideal and $M$ is a maximal Cohen-Macaulay $A$-module then we show $e_i^I(M) = 0$ for $2 \leq i \leq d$. We also have $G_I(M)$ is generalized Cohen-Macaulay. Furthermore we show that there exists $c_I$ (depending only on $A$ and $I$) such that $\text{reg} \ G_I(M) \leq c_I$.