Admissible-solution parity conjecture for the convolution equation

Let $n$ be an integer and let an admissible solution mean a solution of the convolution equation defined earlier in the paper satisfying the stated admissibility conditions. Parity conjecture. (a) There is exactly one real-valued admissible solution when $n$ is even, namely

$$\widehat{\psi}=\frac{\widehat{\varphi}_{n+2}}{\|\varphi_{n+2}\|_{n+2}^{1+1/(n+1)}}.$$

(b) There are no real-valued admissible solutions when $n$ is odd. The preceding discussion gives evidence for the even/odd distinction, including an obstruction to real-valued even solutions in the odd case.