2-adic valuation conjecture when k has one binary digit

Let $d_2(k)$ be the sum of the binary digits of $k$, let $a=\operatorname{ord}_2(m)$, and let $o=\operatorname{ord}_2(A_{\overline{\mathcal K},k,m}(0))$. Binary-digit valuation conjecture. If $d_2(k)=1$ and $a\geq2$, then

$$o=k(a+2)+3.$$

This belongs to the source's collection of empirical conjectures for parameters other than the prime-power cases; its status is unresolved in the supplied text.