Generalized unbounded improvement conjecture for $(k,\ell)$-sum-free sets

Let $k<\ell$ be positive integers. For a set $A$ of $N$ positive integers, let $\mathscr{M}_{(k,\ell)}(A)$ be the size of its largest $(k,\ell)$-sum-free subset, where a set is $(k,\ell)$-sum-free if, for every $k+\ell$ elements $x_1,\dots,x_k,y_1,\dots,y_\ell$, one has

$$\sum_{i=1}^{k}x_i\ne\sum_{j=1}^{\ell}y_j.$$

Define

$$\mathscr{M}_{(k,\ell)}(N)=\inf_{A\subseteq\mathbb{Z}^{>0},\ |A|=N}\mathscr{M}_{(k,\ell)}(A).$$

A generalized $(k,\ell)$-sum-free conjecture. There is a function $\omega(N)\to\infty$ as $N\to\infty$ such that for every set $A$ of $N$ positive integers, there exists a maximal $(k,\ell)$-sum-free set $\Omega(k,\ell)\subseteq\mathbb{R}/\mathbb{Z}$ for which

$$\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}\left(\mathbbm{1}_{\Omega(k,\ell)}-\frac{1}{k+\ell}\right)(nx)=\omega(N).$$

The known asymptotic formula $\mathscr{M}_{(k,\ell)}(N)=N/(k+\ell)+o(N)$ leaves open whether every pair $(k,\ell)$ admits an unbounded improvement over the probabilistic baseline $N/(k+\ell)$.