Uniform bounded geometry conjecture for hyperbolic components

Let $\mathcal{M}$ be the Mandelbrot set, let $\mathcal{H}\subset\mathcal{M}$ be a hyperbolic component, and let $\mathcal{M}_{\mathcal{H}}$ be the associated small copy of $\mathcal{M}$ centered at $\mathcal{H}$. Write $\operatorname{diam}$ for diameter and call a disk or filled cardioid uniformly quasiconformal if its quasiconformal geometry is controlled independently of the component. Bounded geometry conjecture. Every closed hyperbolic component $\overline{\mathcal{H}}$ is a uniform quasiconformal disk or a uniform quasiconformal filled cardioid; its diameter $\operatorname{diam}(\mathcal{H})$ is uniformly comparable to $\operatorname{diam}(\mathcal{M}_{\mathcal{H}})$. This predicts uniform geometric control of hyperbolic components and their associated Mandelbrot copies; the source does not state a resolution.