The full ambient conormal is still the relevant one

Locally choose further coordinates $(\tau ,\eta )$ on $S_{\mathrm{resp}}$ with $N_U=\{\eta =0,n=0\}$ and $\eta \in {\mathbb{R}}^3$. If a coefficient extension were conormal to $N_U$ inside $S_{\mathrm{resp}}$, its insertion would have covectors

These are precisely full ambient conormal covectors of $N_U$ in $U$.

Thus the normal-jet interface retains all eight ambient normal directions. It does not discard the attachment directions, and it assigns no degree from the relative three-dimensional notation.

No conormal landing condition or uniform estimate for the coefficient extensions is proved here.