Strong Nagata conjecture for positive-genus curves

Let $X_n$ be the projective plane blown up at $n$ general points, and let $C$ be an irreducible curve of genus $g>0$ on $X_n$. Let $(3;1^9,0^{n-9})$ denote the indicated plane linear-system class, and let $(CK)$-equivalence mean equivalence under the Cremona–isometry action. Strong Nagata conjecture. One has $C^2>0$, except when $n\geq9$, $g=1$, and $C$ is $(CK)$-equivalent to $(3;1^9,0^{n-9})$; in that exceptional case $C^2=0$. The source describes this as plausible and notes that it would follow from the strong negativity conjecture.