The asymptotic density conjecture for $(p,q)$-sum-free subsets of the square grid

Let $p,q$ be positive integers. A set $S\subseteq[n]^2$ is $(p,q)$-sum-free if the equation $px+qy=z$ has no solution with $x,y,z\in S$, and define

$$\mu_{[p,q]}([n]^2):=\max\left\{\frac{|S|}{n^2}\mid S\subseteq[n]^2\text{ is $(p,q)$-sum-free}\right\}.$$

Asymptotic density conjecture. For positive integers $p,q$ and sufficiently large $n$,

$$\mu_{[p,q]}([n]^2)=\left(1-\frac{2}{(p+q)^2+1}\right)+O(1/n).$$

The formula matches the density supplied by the stripe construction. The paper gives an upper bound of this form in the special case $p=q$, while the corresponding assertion for general positive integers $p,q$ remains conjectural.