Flatness conjecture for high-rank polynomial pullback maps

Let $k$ be a field, let $\bb V$ and $\bb W$ be $k$-vector spaces with $\dim(\bb W)\leq m$, and let $P:\bb V\to k$ be a polynomial of degree $d$. For an affine map $\phi:\bb W\to\bb V$, write $\kappa_P(\phi)=P\circ\phi$, viewed as a point of the variety of polynomials of degree at most $d$ on $\bb W$. Flatness conjecture. For any $m,d\geq 0$, there exists $\rho=\rho(m,d)$ such that if the rank of $P$ is at least $\rho(m,d)$, then the map $\kappa_P$ is flat. Non-archimedean local fields are also expected to have property $c_\infty$, and a polynomial bound for the rank in the complete-intersection result is expected to follow from the bias-rank conjecture, known for $d=2,3$.