The repelling 45-lines conjecture for the sextic map $h_{19}$

Let $h_{19}$ be the dynamical map on ${\mathbf{CP}^{2}}$, let $J_{h_{19}}$ denote its Julia set, and let the $45$-lines be the invariant lines represented by

$$\{X=0\}.$$

Repelling 45-lines conjecture. On the $45$-lines $h_{19}$ is repelling and, hence, $\{X=0\}$ resides in the Julia set $J_{h_{19}}$.

The claim is motivated by basin plots showing repelling behavior along the real projective lines where the real projective planes meet invariant $45$-lines. The supplied text gives no evidence that the conjecture has been resolved.