= Solution
On the positive $z$ axis, $x=y=0$ and $r=z$. Taking this limit in the spatial components gives
$$
h_{xx}=-A+2C,\qquad
h_{yy}=A-2C,\qquad
h_{xy}=h_{xz}=h_{yz}=h_{zz}=0,
$$
with $A,B,C$ evaluated at retarded time $t-r$. A wave propagating toward the observer along the $z$ direction has polarization matrix
$$
h_{ij}^{\rm TT}
=
\begin{pmatrix}
h_+&h_\times&0\\
h_\times&-h_+&0\\
0&0&0
\end{pmatrix}.
$$
Consequently the observed <gravitational wave polarization> amplitudes are
$$
\boxed{h_+=-A+2C,\qquad h_\times=0}.
$$
The $B$ mode and the orthogonal combination of $A,C$ do not contribute on this symmetry axis. The observer therefore sees a purely plus-polarized wave; a rotation of the transverse $x,y$ axes would represent the same physical polarization as the corresponding spin-two mixture of plus and cross.
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