The conjecture that the reconstruction gap is unbounded

Let $G(n)$ denote the largest integer $k$ such that every partition of $n$ is uniquely determined by its multiset of $k$-minors, with the precise definition of $G(n)$ as in the paper. The quantity $n-G(n)$ measures the gap between the size of the partition and the largest reconstructible minor size. Unbounded reconstruction-gap conjecture. The difference $n-G(n)$ satisfies

$$\lim_{n \rightarrow \infty} n-G(n) = +\infty.$$

The paper proves that $n-G(n)=O(n/\log n)$, while computational results determine this difference for various values of $n$. The conjecture asserts that the gap is nevertheless unbounded.