Uniform shifted-sum estimate for the discrete null cone

Let $k$, $L$, $\z_0\in\mathbb{R}^k$, and $f$ be as in the shifted sum

$$S_L^{z_0}(f):=\sum_{z\in\mathbb{Z}_L^k\cap\Sigma}f(z-z_0).$$

Uniform shifted-sum conjecture. The shifted sum satisfies estimate uniformly in $z_0$. Accordingly, for $d\geq 3$ and $a,b\in\mathbb{R}^d$, the sum

$$\sum_{u,v\in\mathbb{Z}_L^d:\,u\cdot v=0}\frac{1}{\langle(u+a,v+b)\rangle^\mu}$$

is bounded by $C_dL^{2(d-1)}$ uniformly in $a,b$.

The claim strengthens the preceding shifted-sum and weighted null-cone estimates by removing the factors depending on the shifts. The supplied text does not establish the uniform estimate or provide evidence of its resolution.