A reduction theorem for Lê's conjecture

Let $\mathit{n}=(n_1,n_2,n_3):({\mathbb{C}}^2,0)\to ({\mathbb{C}}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\mathit{n}{)}_0=0$, then ${\mathrm{o}\mathrm{r}\mathrm{d}}_0(\mathit{n})\in \{2,3,4\}$. This reduces Lê’s conjecture to excluding potential counterexamples $\mathit{n}$ of orders $2$, $3$, and $4$. The proof combines techniques from the theory of plane curve singularities, explicit cobordism constructions, and genus bounds obtained by applying properties of the $\mathrm{{\rm Y}}$ invariant coming from knot Floer homology.