A regulator-sum bound for Scholz's conjecture

Let $n\geq 3$, and let $1,2,\ldots,s_{\iota(n)-1},s_{\iota(n)}$ be a shortest addition chain producing $n$ of length $\iota(n)$. Write its associated generators as

$$2,s_3=a_3+r_3,\ldots,s_{\iota(n)}=a_{\iota(n)}+r_{\iota(n)},s_{\iota(n)+1}=a_{\iota(n)+1}+r_{\iota(n)+1}=n.$$

Scholz's conjecture reduction. The following inequality holds:

$$\iota(2^n-1)\leq \bigg(\sum \limits_{j=2}^{\iota(n)}a_j\bigg)+r_{\iota(n)}+r_{\iota(n)+1}+1-\int \limits_{2}^{\iota(n)-1}\sum \limits_{2\leq j\leq t}r_jdt.$$

This is presented as a reduction of Scholz's conjecture using the identity for addition-chain lengths, rather than as a separate conjecture; the source does not provide a resolution of the underlying Scholz inequality.