Bi-Lipschitz equivalence conjecture at the boundary of extra-nice dimensions

Let $(n,p)$ lie on the boundary of the extra-nice dimensions. Consider generic immersions

$$g,g':(\mathbb C^{p},0)\to(\mathbb C^{p+1},0).$$

Suppose these immersions define germs

$$f,f':(\mathbb C^{n},0)\to(\mathbb C^{p},0).$$

Two immersions are bi-Lipschitz $\mathcal{K}_{V}$-equivalent when they are equivalent under the corresponding bi-Lipschitz contact equivalence preserving the relevant variety $V$.

Boundary extra-nice-dimensions conjecture. Any two generic immersions $g$ and $g'$ are bi-Lipschitz $\mathcal{K}_{V}$-equivalent, and the germs $f$ and $f'$ they define are bi-Lipschitz $\mathcal{A}$-equivalent.

The source notes that the relevant families are topologically trivial, while Whitney equisingularity and bi-Lipschitz triviality remain open at the boundary of the extra-nice dimensions. The conjecture asks for the corresponding uniform bi-Lipschitz classification.