Carleson's $\varepsilon^2$-conjecture for tangent points of Jordan domains

Let $\Omega\subset\mathbb{R}^2$ be a Jordan domain, and let $\partial\Omega$ denote its boundary. For $x\in\partial\Omega$, let $\varepsilon(x,r)$ be the Carleson $\varepsilon$-function, which measures how much the two largest arcs of $\partial B(x,r)$ contained in the complementary components of $\Omega$ differ in length from $\pi r$. A point $x$ is a tangent point of $\partial\Omega$ when the boundary has a tangent at $x$. Carleson's $\varepsilon^2$-conjecture. Except for a set of zero $\mathcal{H}^1$-measure, $x$ is a tangent point of $\partial\Omega$ if and only if

$$\int_0^1 \varepsilon(x,r)^2\frac{dr}{r}<\infty.$$

The conjecture proposes a Dini-type characterization of boundary tangency through quantitative control of the Carleson $\varepsilon$-function for Jordan domains. Its resolution status is not established by the supplied text.