= Solution
Assume a complete relativistically normalized basis of scalar <momentum eigenstates> transforms as
$$
\hat T|\boldsymbol p\rangle=e^{i\chi(\boldsymbol p)}|-\boldsymbol p\rangle,\qquad \chi(\boldsymbol p)\in\mathbb R.
$$
This states both reversal of spatial <momentum> and preservation of the normalization of basis <quantum states>. <Antilinearity> alone would not suffice: multiplying a conjugation operator by two gives an <antilinear operator> that multiplies squared <norms> by four.
Expand $|\phi\rangle=\int d\Pi_p\,\phi(\boldsymbol p)|\boldsymbol p\rangle$ and similarly for $|\psi\rangle$, with $d\Pi_p=d^3p/((2\pi)^3\,2E_p)$. This <Lorentz-invariant phase-space measure> is unchanged under $\boldsymbol p\mapsto-\boldsymbol p$. The <antilinearity> of the <quantum time-reversal operator> gives conjugated expansion coefficients. <Orthogonality> of the <momentum eigenstates> and cancellation of their unit-modulus phases yield
$$
\langle\hat T\phi|\hat T\psi\rangle=\int d\Pi_p\,\phi(\boldsymbol p)\psi(\boldsymbol p)^*=\langle\phi|\psi\rangle^*.
$$
Since momentum reversal is a bijection of the complete basis, $\hat T$ is also onto. Thus
$$
\boxed{\langle\hat T\phi|\hat T\psi\rangle=\langle\phi|\psi\rangle^*,\qquad \hat T\text{ is antiunitary}.}
$$
This proves <antiunitarity>, with the normalization hypothesis explicitly included. For scalar multiparticle <quantum states> the same argument uses the complete occupation-state basis and reverses all momenta; it is not restricted to a single-particle <wave packet>.
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