The defining-equations conjecture for the odeco variety

Let $S^d(\mathbb C^n)$ be the space of symmetric tensors, and let the odeco variety be the Zariski closure of the orthogonally decomposable tensors. For a tensor $T\in S^d(\mathbb C^n)$, write its associated homogeneous polynomial as

$$f_T(x_1,\dots,x_n)=\sum_{i_1+\cdots+i_n=d}\frac{1}{i_1!\cdots i_n!}u_{i_1,\dots,i_n}x_1^{i_1}\cdots x_n^{i_n},$$

where $u_{i_1,\dots,i_n}=d!T_{\scriptsize{\underbrace{1\dots 1}_{i_1\text{ times}}\dots\underbrace{n\dots n}_{i_n\text{ times}}}}$ and $e_s$ is the $s$th standard basis vector of $\mathbb Z^n$. The defining-equations conjecture. The prime ideal of the odeco variety inside $S^d(\mathbb C^n)$ is generated by the quadratic equations

$$\sum_{s=1}^n u_{y+e_s}u_{v+e_s}-u_{w+e_s}u_{z+e_s}=0,$$

where $y,v,w,z\in\mathbb Z_{\geq 0}^n$ satisfy

$$\sum_i y_i=\sum_i v_i=\sum_i z_i=\sum_i w_i=d-1$$

and $y+v=z+w$. This would give an explicit set of defining equations for the variety of orthogonally decomposable symmetric tensors; the result is proved in the paper for $n=2$, while the general assertion remains open.