The real-slice conjecture for generalized Mandelbrot sets of even degree

Let ${\mathcal{M}^p}$ be the generalized Mandelbrot set for the polynomial $Q_{p,c}(z)=z^p+c$, where $z,c\in\mathbb{C}$ and $p\geq 2$ is an integer. The real-slice conjecture. If $p$ is even, then

$${\mathcal{M}^p}\cap\mathbb{R}=\left[-2^{\frac{1}{p-1}},(p-1)p^{\frac{-p}{p-1}}\right].$$

The conjecture is proposed as the missing even-degree analogue of the paper's result for Multibrot sets of odd degree; proving it would complete the characterization of the real intersections needed for the even-power case.