= Solution
Contract the numerator of the quoted <light cone gauge> <propagator> with $n_\mu$:
$$
n_\mu\left[g^{\mu\nu}-\frac{n^\mu p^\nu+n^\nu p^\mu}{n\cdot p}\right]
=n^\nu-\frac{n^2p^\nu+(n\cdot p)n^\nu}{n\cdot p}
=-\frac{n^2p^\nu}{n\cdot p}=0.
$$
Thus every internal <gluon> emerging from a <ghost-gluon vertex> contributes a factor $n_\mu\widetilde\Delta^{\mu\nu}=0$. Every ghost-dependent contribution to a physical <gluon> or matter amplitude vanishes. Equivalently, the strict condition $n\cdot A=0$ removes the <Faddeev-Popov ghost> interaction in <2/b/i>, leaving the field-independent determinant $\det(n\cdot\partial)$, which cancels from normalized correlators.
This is <ghost decoupling in strict axial gauge>, not a statement that the free <Faddeev-Popov ghost> <propagator> itself is zero. The algebra assumes a consistent <axial-gauge pole prescription> for $n\cdot p=0$ and treatment of residual gauge zero modes. The overall sign assigned to the displayed vector <propagator> does not enter the contraction proof. With the mostly-minus convention and the usual $-F^2/4$ action, its conventional prefactor is $-i/(p^2+i0)$; the source prints $+i$. Either prefactor gives the same zero contraction, so no sign correction is needed to obtain the requested ghost-decoupling result.
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