The representation-theoretic Mathieu conjecture for symmetric powers

Let $A=\bigoplus_{n\in {\mathbb{N}}}S^{dn}{\mathbb{C}}^N\otimes S^k({\mathbb{C}}^N)^*$ be a graded $SL(N,{\mathbb{C}})$-algebra. Let $P^+$ denote the set of dominant highest weights, choose $\tau,\mu\in P^+$, and let $f\in A$. For an element of $A$, write $(f^n)_\lambda$ for its projection to the isotypical component indexed by the highest weight $\lambda$. The representation-theoretic Mathieu conjecture. If $(f^n)_{n\tau}=0$ for all $n\in {\mathbb{N}}$, then $(f^n)_{n\tau+\mu}=0$ for all sufficiently large $n$. This is the algebraic representation-theoretic form used in the paper's approach to the Jacobian Conjecture; the source does not specify whether this formulation has been resolved.