The asymptotic restricted $(k,\ell)$-sum-free set conjecture

Let $\widehat{\mathscr{M}}_{(k,\ell)}(N)$ denote the restricted analogue studied in the paper. For positive integers $k$ and $\ell$ satisfying $k>2\ell+1$,

The restricted $(k,\ell)$-sum-free asymptotic conjecture.

$$\widehat{\mathscr{M}}_{(k,\ell)}(N)=\left(\frac{1}{k+\ell}+o(1)\right)N.$$

The paper says that the strategy proving the corresponding theorem fails when $k>2\ell+1$, and that new ingredients may be required; the assertion is therefore open.