The resultant formula for parametrized surfaces with non-local-complete-intersection base points

Let $I \subset \mathbb{C}[s,t,u]$ be generated by $x,y,z,w$ of degree $n$, with $V(I)$ finite. Assume that $I$ is saturated and has the free resolution given by the complex referenced in the source. Let $V_1(A)$ and $V_2(A)$ denote the subschemes of $\mathbb{P}^2$ defined by the $1\times 1$ and $2\times 2$ minors of $A$, respectively. Let $P(X,Y,Z,W)=0$ be the implicit equation of the parametrized surface, let $h$ be the degree of the parametrization, and for each $p\in V_2(A)$ let $\ell_p=0$ be the equation of the plane $H_p$ associated with $p$. For the local ideal at $p$, write $e_p$ for its multiplicity and $d_p$ for its degree.

The resultant formula. If $V_1(A)=\emptyset$ and $V_2(A)\ne\emptyset$, then, up to a nonzero constant,

$$\operatorname{Res}_{\mu_1,\mu_2,\mu_3}({\bf p},{\bf q},{\bf r})=P(X,Y,Z,W)^h\prod_{p\in V_2(A)}\ell_p^{e_p-d_p}.$$

This conjectural formula describes the extraneous factors of the resultant arising from the points of $V_2(A)$, while the factor $P^h$ records the implicit surface with parametrization degree $h$. The source does not provide evidence of a resolution, so the conjecture is treated as open.