Maximal-minor truncation conjecture for Jacobian dual matrices

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, let $I=(f_1,\dots,f_r)$ be a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal of $S$, and let

$$\Phi:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{r-1}$$

be the rational map defined by $f_1,\dots,f_r$. Let $\Theta$ be the Jacobian dual matrix of a presentation matrix of $I$, and write $W$ for the image of $\Phi$. Maximal-minor truncation conjecture. One has

$$I_n(\Theta)=I(W)_{\geq n}.$$

The preceding theorem in the supplied text identifies the saturation of $I_n(\Theta)$ with $I(W)$; this conjecture asserts the stronger equality with the degree-$n$ truncation. No resolution is supplied here.