Conjecture on exact lower-bound functions for addition-chain defects

Let $f^\ell(k)$ and $f^\ell_{\mathrm{st}}(k)$ be the lower-bound functions for ordinary and stable addition-chain defects, and let $f^*(k)$ and $f^*_{\mathrm{st}}(k)$ be the corresponding functions for star chains, as defined earlier in the paper. Lower-bound function conjecture. For every integer $k\ge0$,

$$f^\ell(k)=f^\ell_{\mathrm{st}}(k)=k.$$

For every integer $k\ge0$,

$$f^*(k)=f^*_{\mathrm{st}}(k)=k.$$

These equalities would sharpen the lower bounds developed in the paper; the supplied text presents them as conjectures for future work and gives no resolution.