2-adic valuation conjecture for Ā at k=2 and powers of 2

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term. 2-adic valuation conjecture at $k=2$. For every integer $a\geq2$,

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

This is an empirical special case of the prime-power investigations in the source and is not proved there.