On the covering of $n + ε$ square with $n^2 + 1$ unit squares for $n \geq 4$

In 2006, Alexander Soifer conjectured that one cannot fully cover a square of side length $> n$ with $n^2 + O(1)$ unit squares. For small $n \in \{2, 3\}$, in 2009, Janusz Januszewski proved that it is impossible to fully cover a square of side length $> n$ with exactly $n^2 + 1$ unit squares. Recen