Generic three-dimensional invariant-ring conjecture

Let $G=({\bf Z}/p)^r=\langle e_1,\ldots,e_r\rangle$ and let $V$ be the generic three-dimensional representation over $\mathbf{k}=\mathbf{F}_p(x_{1j},x_{2j}\mid j=1,\ldots,r)$. Define $s=\lceil r/2\rceil$. A SAGBI basis is a set of generators whose leading monomials generate the leading-term algebra. Generic invariant-ring conjecture. The invariant ring $\mathbf{k}[V]^G$ is a complete intersection with embedding dimension $s+3$. Furthermore, there exists a SAGBI basis $\{x,f_1,\ldots,f_{s+1},N_G(z)\}$ satisfying the stated leading-monomial and subduction conditions: if $r=2s$, then $\operatorname{LM}(f_1)=y^{p^s}$ and $\operatorname{LM}(f_i)=y^{p^{s+i-2}+2p^{s-i+1}}$ for $i>1$, with relations obtained from the specified pairs; if $r=2s-1$, then $\operatorname{LM}(f_1)=y^{2p^{s-1}}$, $\operatorname{LM}(f_2)=y^{p^s}$, and $\operatorname{LM}(f_i)=y^{p^{s+i-3}+2p^{s-i+1}}$ for $i>2$, with relations obtained from the specified pairs. These assertions are supported by computer calculations in the paper, while the arbitrary-rank claim remains open.