The inherited-arc conjecture for tangent conics in André planes of order q^3

Let ${\textup{PG}}(2,q^3)$ be a projective plane with $q$ even and $q>2$. Let $\mathcal{C}$ be a conic with nucleus $N$ tangent to the line $\ell_\infty$ at $T$, and let $\mathcal{D}$ be an André set of $\ell_\infty$ with transversal points $E,F$. For the André replacement net $\mathcal{D}^m$, write $\mathcal{D}_m(\mathcal{C})$ for the set corresponding to the affine points of $\mathcal{C}$. Inherited-arc conjecture. If

$$|\{T,N\}\cap\{E,F\}|<2,$$

then $\mathcal{D}_m(\mathcal{C})$ is not an arc in the André plane $\mathcal{P}(\mathcal{D}^m)$ for either $m\in\{1,2\}$. Computer searches for $t=3$ and small $q$ found no counterexamples, while the complementary case in which both $T$ and $N$ are transversal points yields inherited arcs; the conjecture asserts that no further cases occur.