Logarithmic asymptotics conjecture for Welschinger invariants of blown-up planes

Assume that $\mathbb{P}^2_k$ is obtained from $\mathbb{P}^2$ by blowing up $k$ generic real points and is equipped with its natural real structure. Let $D\subset \mathbb{P}^2_k$ be a real ample divisor. Logarithmic asymptotics conjecture. The Welschinger invariants $W_{nD}(\mathbb{P}^2_k)$ satisfy

$$\lim_{n\to +\infty} \frac{\log W_{nD}(\mathbb{P}^2_k)}{n\log n}=D\cdot c_1(\mathbb{P}^2_k).$$

This conjecture extends the corresponding asymptotic relation known for genus-zero Gromov–Witten invariants and the cases $k=1,2,3$, where the same relation for Welschinger invariants had been established. The general case for blow-ups at generic real points remains open.