The conjecture on Waring numbers of ramified 2-adic rings

Let $k$ be an even positive integer and let $e\geq 1$. Write $g_{2,e}(k)$ for the least number of $k$-th powers needed to represent every element of the ramified $2$-adic ring of ramification index $e$, and let $g_{\mathbb{Z}_2}(k)=g_{2,1}(k)$ denote the corresponding Waring number over $\mathbb{Z}_2$. The conjecture. If

$$\gcd(e,k)>1,$$

then

$$g_{2,e}(k)<g_{\mathbb{Z}_2}(k);$$

otherwise,

$$g_{2,e}(k)=g_{2,1}(k).$$

This conjecture summarizes the cases calculated in the paper for even $k$ and $p=2$; the general behavior is expected to depend on the relative sizes of $\nu_2(e)$, $\nu_2(k)$, and $k$, but the asserted dichotomy is not established in the supplied text.