Rational embeddability conjecture for circular spherical divisors

Let $D_n$ be the circular spherical divisor with $n$ components discussed above. A rational embedding means an embedding of $D_n$ into a rational surface. Rational embeddability conjecture. $D_n$ is rationally embeddable if and only if $D_n$ is anti-canonical. This would characterize anti-canonical circular spherical divisors by rational embeddability; the source notes that all anti-canonical $D_n$ embed in $\mathbb{C}\mathbb{P}^2\# 9\overline{\mathbb{C}\mathbb{P}}^2$, while the converse is proposed because the only known obstruction rules out embeddings into $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}}^2$ for $k<9$.