Conjecture on recovering double-point structure from the scheme $X_2$

Let $C\subset\mathbb{P}^2$ be a rational plane curve with parameterization ${\bf f}$, and let $X_2$ be the scheme defined in Section 2. Assume that $C$ has only double points as singularities, write

$$\operatorname{Sing}(C)=\{P_1,\dots,P_t\},$$

and let $(X_2)_1,\dots,(X_2)_t$ be the corresponding subschemes of $X_2$. For each $i$, set

$$L_i:=p_2p_1^{-1}((X_2)_i).$$

The $X_2$ multiple-line conjecture. For every $i$, $(X_2)_i$ is curvilinear and $\operatorname{length}(X_2)_i=\delta_{P_i}$. If $P_i$ is an $A_{2m_i-1}$ singularity, then $L_i$ cuts two curvilinear schemes of length $m_i$ on $C_n$, whose images under $\pi$ form a curvilinear subscheme of length $m_i$ of $C$. If $P_i$ is an $A_{2m_i}$ singularity, then $L_i$ is tangent to $C_n$ and cuts a curvilinear scheme of length $2m_i$ on $C_n$, whose image under $\pi$ is a curvilinear subscheme of length $m_i$ of $C$.

The conjecture would make it possible to determine the structure and plane coordinates of the double points from $X_2$ and their preimages on $\mathbb{P}^1$; equivalently, an $A_s$ singularity should be described by $p_2p_1^{-1}(X_2)\cap C_n$ and its projection to $C$.