The strong-obstruction conjecture for the permanent versus determinant

Let $g=\operatorname{det}(Y)$, $h=\operatorname{perm}(X)$, and $f=\phi(h)$. A strong obstruction is a representation-theoretic obstruction of the type defined in the paper for the pair $(f,g)$. Strong-obstruction conjecture. There exist infinitely many strong obstructions for $(f,g)$ if

$$m=2^{\log^c n},$$

where $c$ is a constant and $n\to\infty$. The conjecture would imply $f\notin\Delta_V[g]$, hence the arithmetic implication of the $P^{\#P}\ne NC$ conjecture; it depends on the proposed equations for the determinant orbit closure.