The Sylvester-form conjecture for the Rees ideal of a linearly presented ideal

Let $k[\mathbf{x}]=k[x_1,x_2,x_3]$ and $k[\mathbf{y}]=k[y_1,y_2,y_3,y_4]$. Consider the matrix $\varphi:=\varphi_{3,r}$ as in the paper's template, and let $I:=I_3(\varphi)\subset k[\mathbf{x}]$. Assume that the $2$-minors of the linear part of $\varphi$ generate a $(\mathbf{x})$-primary ideal. For $i=1,2,\ldots,r$, let $f_i$ be the Sylvester form of $\{l_1,l_2,f_{i-1}\}$ with respect to $\{\mathbf{x}\}$, where $l_1,l_2,f_0$ are the forms generating $I_1((\mathbf{y})\cdot\varphi)$. The Rees-ideal conjecture. The ideal $\left(I_1\left((\mathbf{y})\cdot\varphi\right),f_1,f_2,\ldots,f_r\right)$ is the Rees ideal of $I$. The preceding proposition establishes that the Rees ideal is a minimal prime of this ideal and that $f_i$ has bidegree $(r-i,2i+1)$; the conjecture asserts equality, giving the full defining ideal of the Rees algebra in this setting.