The conjecture that unimodular lattices attain secrecy gain at y=1

Let $\Lambda$ be an $n$-dimensional unimodular lattice, and let $\Xi_{\Lambda}(y)$ be its secrecy function for $y>0$. The secrecy gain is $\chi_{\Lambda}=\sup_{y>0}\Xi_{\Lambda}(y)$. Unimodular-lattice secrecy-gain conjecture. The secrecy gain of $\Lambda$ is achieved at $y=1$, namely

$$\chi_{\Lambda}=\Xi_{\Lambda}(1).$$

The symmetry $\Xi_{\Lambda}(y)=\Xi_{\Lambda}(1/y)$ shows that $y=1$ is a natural candidate, but symmetry alone does not establish that it is a global maximizer. The claim concerns the optimization of secrecy for unimodular lattices in the Gaussian wiretap channel.