Elvang and Huang sign error? Polarization Vectors in the Spinor Helicity Formalism

I am reading through Elvang and Huang’s treatment of polarization vectors for all outgoing spin-1 massless particles (metric signature $(-,+,+,+)$).

It is given in Eq. (2.50) in the PDF (but Eq. (2.51) in the textbook version) that the polarization for a particle of momentum $p$, reference spinor $q neq p$, can be written as

begin{equation}tag{1}
not varepsilon_{+}(p ; q)=frac{sqrt{2}}{langle q prangle}(|p]langle q|+| qrangle[p |)
end{equation}

for plus-helicity, and

begin{equation}tag{2}
not varepsilon_{-}(p ; q)=frac{sqrt{2}}{[q p]}(|prangle[q|+| q]langle p|)
end{equation}

for minus helicity.

However, my understanding is that polarization vectors of opposite helicity should be related by complex conjugation; namely, $left(varepsilon_{+}right)^{*}=varepsilon_{-}$.

I don’t see how $(1)$ and $(2)$ satisfy this property. The authors give that $[p|=(|prangle)^{*}$ and $leftlangleleft. pright|=(| p]right)^{*}$, as well as the antisymmetry property $langle p qrangle=-langle q prangle, [p q]=-[q p]$. It therefore seems to me that one of $(1)$ and $(2)$ should have a minus sign out front to satisfy the property $left(varepsilon_{+}right)^{*}=varepsilon_{-}$.

I suspect that things are more complicated because the polarization vectors in $(1)$ and $(2)$ are contracted with $gamma^mu$, but I don’t know of a simple relationship for $(gamma^mu)^*$.