The elimination ideal conjecture for rank-five subgroups containing an $\mathbb{F}_{p^2}$-subspace

Let $p$ be prime, let $d_{i,4}$ and $d_{i,5}$ be the invariant functions associated with rank $4$ and rank $5$ subgroups, respectively, and regard $\mathbb{F}[d_{1,5},\ldots,d_{5,5}]$ as a subalgebra of the polynomial algebra in the rank-$4$ invariants. The ideal intersection

$$\langle d_{1,4},d_{3,4}\rangle\cap\mathbb{F}[d_{1,5},\ldots,d_{5,5}]$$

is the elimination ideal describing the corresponding variety.

Elimination ideal conjecture. One has

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

If true, this gives the proposed equations for the variety of rank $5$ subgroups containing a rank $4$ subgroup that is a vector space over $\mathbb{F}_{p^2}$. The equality was verified for primes less than or equal to $17$.