Modular Rado boundedness conjecture

For $k\in\mathbb{N}$, let $a_1,\ldots,a_k\in\mathbb{Z}/(2^r\mathbb{Z})$, and let $d\leq r$ be the largest non-negative integer such that $2^d$ divides $\sum_{i\in I}a_i$ for some non-empty $I\subseteq[k]$. Modular Rado boundedness conjecture. There exists $K\in\mathbb{N}$, depending only on $k$, such that one can $K$-colour $\mathbb{Z}/(2^r\mathbb{Z})$ so that every monochromatic solution $x=(x_1,\ldots,x_k)$ of

$$a_1x_1+\cdots+a_kx_k=0$$

satisfies $2^{r-d}\mid x_i$ for every $i\in[k]$. The source notes that a compactness argument would imply the single-equation case of Rado's boundedness conjecture.