Aliev's generalized-hexagon conjecture for the critical determinant

For $n\geq3$, let $c(n)$ be the optimal constant defined by

$$c(n)=\sup_{\boldsymbol a\in\mathbb Z^n\setminus\{0\}}\inf_{\substack{\boldsymbol x\in\mathbb Z^n\setminus\{0\}\langle\boldsymbol a,\boldsymbol x\rangle=0}}\frac{\|\boldsymbol x\|_\infty^{n-1}}{\|\boldsymbol a\|_\infty},$$

and let $\Delta(K)$ denote the critical determinant. Let $H_{1,\ldots,1}$ be the generalized hexagon in $\mathbb R^{n-1}$ defined in the source. Aliev's conjecture.

$$c(n)=\Delta(H_{1,\ldots,1})^{-1}.$$

Here $H_{1,\ldots,1}$ is the generalized hexagon in $\mathbb R^{n-1}$; the source states that exact values of $c(n)$ remain unknown for $n>4$.