Isometry classification conjecture for essential-rank-metric spaces

Let $d53d$ be a field, let $S_{n,d}(d53d)$ denote the space of homogeneous degree-$d$ polynomials in $n$ variables, and let $\operatorname{d}_{\operatorname{ess}}$ be the essential-rank metric on this space. For $A\in\operatorname{GL}(n,d53d)$ and $\lambda\ind53d^*$, define the map $f\mapsto\lambda f\cdot A$, where $f\cdot A=f(x\cdot A)$. Isometry classification conjecture. Assume that $\operatorname{char}(d53d)=0$ or $\operatorname{char}(d53d)>d$. Then every isometry of $(S_{n,d}(d53d),\operatorname{d}_{\operatorname{ess}})$ is of the form $f\mapsto\lambda f\cdot A$ for some $A\in\operatorname{GL}(n,d53d)$ and $\lambda\ind53d^*$. The claim aims to classify all linear symmetries preserving essential rank; the preceding discussion establishes that maps of the displayed form are isometries, while the asserted converse remains conjectural under the stated characteristic condition.