  Elina Robeva discovered quadratic equations satisfied by orthogonally
decomposable ("odeco") tensors. Boralevi-Draisma-Horobe\c{t}-Robeva then proved
that, over the real numbers, these equations characterise odeco tensors. This
raises the question to what extent they also characterise the Zariski-closure
of the set of odeco tensors over the complex numbers. In the current paper we
restrict ourselves to symmetric tensors of order three, i.e., of format $n
\times n \times n$. By providing an explicit counterexample to one of Robeva's
conjectures, we show that for $n \geq 12$, these equations do not suffice.
Furthermore, in the open subset where the linear span of the slices of the
tensor contains an invertible matrix, we show that Robeva's equations cut out
the limits of odeco tensors for dimension $n \leq 13$, and not for $n \geq 14$
on. To this end, we show that Robeva's equations essentially capture the
Gorenstein locus in the Hilbert scheme of $n$ points and we use work by
Casnati-Jelisiejew-Notari on the (ir)reducibility of this locus.
