The square characterization conjecture for Hyperbrots of even degree

Let ${\mathcal{D}}$ denote the hyperbolic number plane, and let ${\mathcal{H}}^p$ be the Hyperbrot set associated with the polynomial of degree $p$. Define

$$t_p:=\frac{(1-(2p)^{1/(p-1)})p-1}{2p(p^{1/(p-1)})},\qquad l_p:=\frac{((2p)^{1/(p-1)}+1)p-1}{p(p^{1/(p-1)})}.$$

The square characterization conjecture. For every even integer $p\geq 2$,

$${\mathcal{H}}^p=\left\{x+y\mathbf{j}\in\mathbb{D}:|x-t_p|+|y|\leq\frac{l_p}{2}\right\}.$$

The claim predicts that the even-degree Hyperbrot sets are squares, extending the regular-shape phenomenon established in the paper for the relevant odd-degree systems. The supplied text gives no evidence that this characterization has been proved or disproved.