Noncriticality conjecture for the Hausdorff dimension on the Mandelbrot main cardioid

Let $\mathcal{H}$ be the main cardioid of the Mandelbrot set, and let $\delta: \mathcal{H} \to [0,2)$ be the Hausdorff dimension function. Noncriticality conjecture. The function $\delta$ has no critical point on $\mathcal{H}$ except at $c=0$. This conjecture concerns the variation of the Hausdorff dimension of Julia sets within the main cardioid; the source presents it as an ultimate goal to prove or disprove, and gives no resolution.