Uniqueness of sum-of-squares decompositions up to orthogonal equivalence

Let $f\in \operatorname{Sym}^{2d}V$ be generic of SOS-rank $k\leq n$, and let

$$N=\binom{n+d}{d}.$$

For a decomposition of $f$ into $k$ squares, write $A\in\mathbb C^k\otimes\mathbb C^N$ for the coefficient matrix and let $W_0+C_0$ be its Gram matrix, where $x(W_0+C_0)x^t=f$. The space of such decompositions is Uniqueness conjecture.

$$\mathrm{SOS}_k(f)\cong\\{A\in\mathbb C^k\otimes\mathbb C^N\mid A^tA=W,\\ xWx^t=f\\}=\\{A\in\mathbb C^k\otimes\mathbb C^N\mid A^tA=W_0+C_0\\}=\mathrm O(k).$$

Thus, a generic form of SOS-rank at most $n$ has only one orthogonal orbit of minimal sum-of-squares decompositions. The preceding discussion establishes this behavior for the cases $k\leq 6$ listed in the paper's table; the conjecture asserts it for all generic forms with $k\leq n$.