= Solution
Varying the two-form action gives the gauge-invariant equation
$$
\boxed{\nabla_aF^{abc}=0,\qquad F=dA.}
$$
Choose radial gauge $A_{zi}=0$ and boundary-transverse gauge $\partial^iA_{ij}=0$. In the Poincare patch of $\operatorname{AdS}_6$, $\sqrt{-g}=z^{-6}$, while raising the three indices of $F$ contributes $z^6$. The boundary components consequently obey
$$
\boxed{(\partial_z^2+\eta^{kl}\partial_k\partial_l)A_{ij}=0.}
$$
The near-boundary indicial equation is $\alpha(\alpha-1)=0$, so
$$
A_{ij}(z,x)=J_{ij}(x)+zA^{(1)}_{ij}(x)+\cdots.
$$
Under the bulk dilation $(z,x)\mapsto(\lambda z,\lambda x)$, a two-form component carries two powers of inverse length. Thus the leading coefficient has dimension $2$, while the coefficient of $z$ has dimension $3$:
$$
\boxed{\Delta_J=2,\qquad\Delta_O=3.}
$$
The leading freely specifiable, nonnormalizable coefficient is the CFT source $J_{ij}$. The subleading normalizable coefficient is its canonical response and determines $\langle O_{ij}\rangle$. This agrees with the saturated two-form unitarity bound $\Delta_O=d-2=3$ in the five-dimensional boundary CFT and with the <holographic dictionary> for a massless bulk gauge field.
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