Does this follow from a version of the Strong Approximation Theorem?

I am hoping someone can help me with this.

Define $\Lambda(x)$ to be the set of all 4-tuples $(A,B,C,D)$; $A,B,C,D \in \mathbb{F}_2[x]$ (where $\mathbb{F}_2$ is the field with 2 elements) such that $A,B,C,D$ together satisfy conditions (i) and (ii) below:

(i) $(A^2+AB +B^2) + x(C^2+CD+ D^2)$ is a nonnegative power of $(x+1)$;

(ii) $A-1, B \equiv 0$ mod $x$.

I would like to know if the following is true, and if so, any place in the literature you can point me:

Let $h(x) \in \mathbb{F}_2[x]$ be an irreducible polynomial of degree 4. For arbitrary nonegative integers $r$ and $l$, let $a,b,c,d \in\mathbb{F}_2[x]$ be such that the following equation $$(a^2+ab+b^2) + (c^2+cd +d^2)x \equiv (x+1)^r\mod(h(x))^l$$ holds. Then there exists $(A,B,C,D) \in \Lambda(x)$ s.t. $$A \equiv a \mod (h(x))^l;$$ $$B \equiv b \mod (h(x))^l;$$ $$C \equiv c \mod (h(x))^l;$$ $$D \equiv d \mod h(x))^l.$$

[Now, letting $r'$ be the integer such that the equation $$A^2+AB+B^2 + x(C^2+CD+D^2) = (x+1)^{r'}$$ is satisfied, I don't care whether $r'$ and $r$ as in the above are the same or not, just whether the above in the shaded box is true. I believe the above would be true if we were to replace $(h(x))^l$ with $g(x)$ where $g(x)$ is an irreducible polynomial in $\mathbb{F}_2[x]$.]

This is also a heavily revised version of a question that I asked a few days ago. I have tried looking Strong Approximation in the literature, I must admit, I do not have much of a background in number theory, so I have trouble following.

• This is the kind of thing that should follow from a form of the strong approximation theorem. I don't know of a good reference for quaternary quadratic forms in characteristic two to check that it holds there. – Felipe Voloch Mar 27 '18 at 1:11
• Any more takes on this? – Mike Apr 2 '18 at 17:18