Nonvanishing remainder conjecture for orthogonal Heisenberg invariants

For each $n\geq 1$, let $R_0$ denote the remainder of the element $D_0$ in the construction of the minimal relation for ${\mathcal H}(n)^{O(n)}$. Nonvanishing remainder conjecture.

$$R_0\neq 0.$$

By the lemma in the source, this is equivalent to the existence of a decoupling relation

$$j^{n^2+3n}=P(j^0,j^2,\dots,j^{n^2+3n-2})$$

in ${\mathcal H}(n)^{O(n)}$; it is therefore another formulation of the decoupling conjecture above rather than a distinct claim.