The Sárközy–Sós conjecture for multivariate linear forms

Let $d\geq 1$ and let $k_1,\dots,k_d$ be positive integers. For an infinite set of positive integers $\mathcal{A}$, write $r_{\mathcal{A}}(n;k_1,\dots,k_d)$ for the number of representations of $n$ by the corresponding multivariate linear form.

Sárközy–Sós conjecture. There exists some infinite set of positive integers $\mathcal{A}$ such that $r_{\mathcal{A}}(n;k_1,\dots,k_d)$ is constant for $n$ large enough if and only if, up to permutation of the indices,

$$(k_1,\dots,k_d)=(1,k,k^2,\dots,k^{d-1})$$

for some $k\geq 2$.

The conjecture extends the known construction of Moser for coefficients $k_i=k^{i-1}$ and concerns the exceptional cases not covered by the paper's nonexistence results, including representation functions such as $r_{\mathcal{A}}(n;1,1,2)$.