Finite-dimensionality conjecture for link invariants

Let $L$ be a link whose components are labeled by representations, and let $\Phi(L)$ denote its invariant. A representation is minuscule in the usual representation-theoretic sense.

Finite-dimensionality conjecture. The invariant $\Phi(L)$ is only finite-dimensional if all components of $L$ are labeled with minuscule representations.

This predicts that infinite-dimensional invariants are typical when at least one component is labeled by a non-minuscule representation; the source gives the claim without resolving its status.