Conjecture on determinantal equations of eigenschemes of symmetric tensors

Let $n\geq 2$. For a homogeneous polynomial $f\in\mathbb{C}[x_0,\dots,x_n]_d$, let $E(f)$ denote its eigenscheme. Consider a set $\{f_{ij}\mid 0\leq i<j\leq n\}\subseteq\mathbb{C}[x_0,\dots,x_n]_d$ of $\binom{n+1}{2}$ homogeneous polynomials. Determinantal equations conjecture. This set is the set of determinantal equations of $E(f)$ for some $f\in\mathbb{C}[x_0,\dots,x_n]_d$ if and only if, for every $0\leq i<j<k\leq n$,

$$x_i f_{jk}-x_j f_{ik}+x_k f_{ij}=0$$

and

$$\partial_i f_{jk}-\partial_j f_{ik}+\partial_k f_{ij}=0.$$

The conjecture proposes an intrinsic characterization of the determinantal equations arising from eigenschemes of symmetric tensors; its status is not resolved in the supplied source context.