Mumford's conjecture on invariant complements in symmetric powers

Let $G$ be a semisimple algebraic group and $V$ a $G$-module. Let $V_0$ be a $G$-invariant submodule of codimension one.

Mumford's conjecture. For some $d\in \mathbb{Z}_{+}$, the submodule

$$V_0\cdot S^{d-1}(V)\subseteq S^d(V)$$

has a $G$-stable invariant complement of dimension one.

This conjecture is presented as the origin of the notion of geometric reductivity; the supplied text does not state whether it has been resolved.