Conjecture that constant image Milnor number implies excellence of unfoldings

Let $f:({\mathbb C}^n,\underline{z})\to({\mathbb C}^p,0)$, with $n<p$, be a corank $1$ complex analytic multi-germ with an origin-preserving one-parameter unfolding. Here, $\mu_I$ denotes the image Milnor number, and an unfolding is excellent when it satisfies the excellence conditions considered in the paper.

Excellence conjecture. If $\mu_I$ is constant, then the unfolding is excellent.

The conjecture proposes that the image Milnor number completely controls excellence in this setting. The preceding results establish this implication in several cases, but the general assertion remains open.