Julia-set containment of elliptic-curve attracting basins

Let $f$ be a rational map of ${\mathbb P}^2({\mathbb C})$, let ${\mathcal C}\subset{\mathbb P}^2({\mathbb C})$ be an $f$-invariant elliptic curve, let ${\mathcal B}({\mathcal C})$ be the set of points whose forward orbits converge to ${\mathcal C}$, and let $J(f)$ be the Julia set of $f$. Julia-basin conjecture. The attracting basin satisfies

$${\mathcal B}({\mathcal C})\subset J(f).$$

The paper motivates this by the expected expansion of $f$ in directions tangent to ${\mathcal C}$; the claim is listed among open problems.