Counting conjecture for sum-free subsets

Let $\mathcal{L}$ denote the equation $x+y=z$. A subset of $[n]$ is $(\mathcal{L},2)$-free if it admits a 2-colouring with no monochromatic solution to $\mathcal{L}$. Counting conjecture. There are

$$\Theta\left(2^{4n/5}\right)$$

$(\mathcal{L},2)$-free subsets of $[n]$. The preceding result gives only $2^{4n/5+o(n)}$ such subsets; the conjecture asks that the error in the exponent be replaced by a constant-factor asymptotic estimate.