Concrete evidence for the asymptotics of $\lambda_1(\Lambda^\perp(A))$?
The first inequality at the bottom of page 3 of the paper is false. For example, Conway and Thompson proved the existence of "self-dual" $n$-dimensional lattices $L$ (i.e., $L^* = L$) where $\lambda_1(L) = \lambda_1(L^*) = \Omega(\sqrt{n})$, hence $\eta_{2^{-n}}(L) = O(1)$ but $\tilde{bl}(L) \geq \lambda_1(L) = \Omega(\sqrt{n})$.

The statement and proof of Conway and Thompson's result can be found as Theorem 9.5 in https://www.springer.com/us/book/9783642883323 .