The unbounded-drop conjecture for integer complexity

For each positive integer $n$, let $\Vert n\Vert$ be the least number of $1$'s needed to represent $n$ using addition and multiplication. The quantity $\Vert n-1\Vert-\Vert n\Vert$ measures the drop in complexity from $n-1$ to $n$. Unbounded-drop conjecture.

$$\limsup_{n\to\infty}\bigl(\Vert n-1\Vert-\Vert n\Vert\bigr)=+\infty.$$

Computations in the range $n\leq 905\,000\,000$ found values through $8$, but no larger ones. The conjecture asserts that such drops nevertheless occur with arbitrarily large magnitude.