The polynomial-field conjecture on real Grassmannians

Let $G_n^k$ be the Grassmannian of linear $k$-subspaces of $\mathbb R^n$, let $\gamma_n^k$ be its canonical vector bundle, and let $\Sigma^d(\gamma_n^k)$ denote the fiberwise symmetric $d$-th power. Let $Q(\gamma_n^k)$ be the fiberwise positive quadratic form induced by a Riemannian metric. Suppose $d$ and $k$ are even positive integers.

The polynomial-field conjecture. There exists $n(d,k)$ such that, for every section of $\Sigma^d(\gamma_n^k)$ over $G_n^k$ with $n\geq n(d,k)$, there exists $V\in G_n^k$ such that the section over $V$ is a multiple of $(Q(\gamma_n^k))^{d/2}$.

This stronger Grassmannian-valued formulation would imply the polynomial Dvoretzky theorem. The source states that the conjecture is false in general, although it is proved there for several cases, including odd $d$ and the complex Grassmannian; hence the general assertion is refuted.