Mathieu's conjecture for finite-type functions on compact connected Lie groups

Let $K$ be a compact connected Lie group, and let $f$ and $h$ be complex-valued finite-type functions on $K$. Let $dk$ denote Haar measure on $K$. Mathieu's conjecture. If

$$\int_K f^n(k)\,dk=0$$

for all $n\in {\mathbb{N}}$, then

$$\int_K f^n(k)h(k)\,dk=0$$

for all sufficiently large $n$. The conjecture connects representation theory of compact Lie groups with the Jacobian Conjecture and is presented as the key conjectural input to Mathieu's approach; its resolution status is not specified in the source.