Alternate normal form conjecture for generic unfoldings of saddle-nodes

Let $k\in\mathbb{N}$, let $\mu\in\germ{\varepsilon}$ be a germ of holomorphic function, and let $\widehat{\tau}\in\mathbb{Z}_{\geq0}$ satisfy $\mu_{0}+\widehat{\tau}\notin\mathbb{R}_{\leq0}$. Consider a generic convergent unfolding of a germ of saddle-node holomorphic vector field with formal invariant $\mu$.

Alternate normal form conjecture. Any such unfolding is orbitally conjugate to an unfolding of the form

\fonf+y\widehat{R}\pp y, \qquad \widehat{R}\in x\pol x_{<k}\left\\{x^{\widehat{\tau}}y\right\\}.

Moreover, this form is unique up to conjugacy by linear maps

$$(\varepsilon,x,y)\mapsto(\varepsilon,x,c_{\varepsilon}y),$$

where $c\in\germ{\varepsilon}^{\times}$. This proposed form is intended as a generalization of earlier normal forms, while the existence and uniqueness assertion is presented in the source as a conjectural extension.