2-adic valuation conjecture for two binary digits and m congruent to 2 modulo 4

Let $d_2(k)$ be the sum of the binary digits of $k$. Two-digit valuation conjecture. If $d_2(k)=2$, $m=4j+2$ for $j=1,2,\ldots$, and $a=\operatorname{ord}_2(m)=1$, then

$$\operatorname{ord}_2\left(A_{\overline{\mathcal K},k,m}(0)\right)=(a+6)k+2=7k+2.$$

This is another empirical conjecture about the 2-adic orders of constant terms for non-prime-power values of $m$; the source supplies no resolution.