Elementary abelian 2-group Lehmer constant conjecture

For an integer $n\geqslant2$, let $\mathbb{Z}_{\!/2}^n$ denote the direct sum of $n$ copies of the cyclic group $\mathbb{Z}_{\!/2}$. Let $\lambda(\mathbb{Z}_{\!/2}^n)$ denote its Lehmer constant. Elementary abelian 2-group Lehmer constant conjecture.

$$\lambda(\mathbb{Z}_{\!/2}^n)=\frac1{2^n}\log(2^n-1)$$

for all $n\geqslant2$. The case $n=2$ is established in the preceding example, where the value is $\frac14\log3$; further numerical work is presented as evidence for the asserted formula in every higher dimension.