Strengthened $2$-adic Zeros Conjecture for the continued-fraction denominators of $e$

Let $\mathbb{Z}_2$ be the $2$-adic integers, let $\widetilde{Q}_i:\mathbb{Z}_2\to\mathbb{Z}_2$ be the continuous extensions of $Q_i(n)=Q(3n+i)$, and let $c_k$ be the unique zero of $A(n)$ modulo $2^k$, with $c$ their $2$-adic limit. Strengthened Zeros Conjecture. For all $n\in\mathbb{Z}_2$ and $k\geq 1$,

$$|\widetilde{Q}_0(n)|_2\geq |4n(n+2)|_2,$$ $$|\widetilde{Q}_1(n)|_2\geq |2(n+1)|_2,$$ $$|\widetilde{Q}_2(n)|_2=1,$$

and

$$|c-c_k|_2\geq 2^{-2k}.$$

This is stated conditionally on the Period Conjecture and is described as a slightly stronger replacement for the original Zeros Conjecture. The supplied text gives no resolution.