Full-basin conjecture for a smooth circle covering with a repelling fixed point

Let $\mathcal{D}$ be the set of circle local homeomorphisms $f:\mathbb{S}^1\to\mathbb{S}^1$ that are topologically conjugate to the doubling map. For $f\in\mathcal{D}$, let $p$ be its unique fixed point, let $\delta_p$ be the Dirac measure at $p$, and write $B_{\delta_p}$ for the basin of $\delta_p$. Full-basin conjecture. There exists

$$f\in\mathcal{D}\cap C^1(\mathbb{S}^1)$$

such that $f'(p)>1$ and

$$|B_{\delta_p}|=1.$$

The conjecture strengthens the paper's $C^1$ construction, which gives a positive-Lebesgue-measure basin for a hyperbolic repelling fixed point but not a full-measure basin. The statement asks whether the basin can have full Lebesgue measure under the same regularity and circle-covering restrictions.