The elimination ideal conjecture for a dilation of $\mathbb{F}_{p^2}$

Let $p$ be prime, let $d_{i,4}$ and $d_{i,5}$ be the invariant functions associated with rank $4$ and rank $5$ subgroups, and set

$$r_1:=d_{5,5}^p-d_{4,5}d_{1,5}^p,\qquad r_2:=d_{3,5}^p-d_{2,5}d_{1,5}^p.$$

The ideal generated by

$$d_{1,4}^{p^2}d_{2,4}^pd_{3,4}-d_{4,4}^pd_{1,4}^{p^2+1}-d_{3,4}^{p^2+1}$$

intersects the rank-$5$ invariant subalgebra in an ideal describing the corresponding variety.

Dilation elimination conjecture. One has

$$\langle d_{1,4}^{p^2}d_{2,4}^pd_{3,4}-d_{4,4}^pd_{1,4}^{p^2+1}-d_{3,4}^{p^2+1}\rangle\cap\mathbb{F}[d_{1,5},\ldots,d_{5,5}] =\langle r_1^{p^2+1}-r_2^{p^2}(d_{5,5}^pd_{2,5}-d_{4,5}d_{3,5}^p)\rangle.$$

The surrounding argument states that membership in the variety on the left implies that $E$ contains a dilation of $\mathbb{F}_{p^2}$. The source does not report a verification range for this equality.