Uniqueness and stabilization conjecture for two-orthogonal tensor decompositions

Let $W_r$ be the set of tensors in $\mathbb{R}^{n_1}\otimes\cdots\otimes\mathbb{R}^{n_d}$ admitting a two-orthogonal decomposition with at most $r$ summands. Let $N$ be

$$N=\min_{k\in[d]}\prod_{j\neq k}n_j.$$

Two-orthogonal uniqueness and stabilization conjecture. A generic two-orthogonal tensor in $\mathbb{R}^{n_1}\otimes\cdots\otimes\mathbb{R}^{n_d}$ has a unique two-orthogonal decomposition, up to reordering of the summands. In particular, the chain

$$W_1\subseteq W_2\subseteq\cdots\subseteq W_r\subseteq\cdots$$

stabilizes exactly at $N$:

$$W_{N-1}\neq W_N=W_{N+1}=\cdots.$$

The conjecture extends the verified uniqueness phenomenon from small cases, including eight summands in $(\mathbb{R}^2)^{\otimes 4}$, and predicts the precise point at which allowing additional summands no longer enlarges the set.