Let $(R,m,k)$ be a complete, reduced, local ring of dimension $1$. Let $I$ be a $m$-primary ideal. Define $L(I):= e(I)/\ell(R/I)$.

Wilf conjecture asserts that $L(\mathfrak{c})\leq \mu(m)$, where $\mathfrak{c}$ is the conductor of the normalization. What about other conductors or colon ideals? In particular, $I_a=x^a:m^a$, where $x$ is a fixed reduction of $m$?