Every line segment intersects the escaping set of the exponential map

Let $E(e^z)=\{z\in\mathbb{C}: e^{z},e^{e^{z}},\ldots,e^{\circ n}(z)\to\infty\}$ be the escaping set of the map $z\mapsto e^z$. Escaping-set intersection conjecture. Every line segment intersects $E(e^z)$. This conjecture concerns the geometric size and distribution of escaping points for the complex exponential map; the source attributes it to Lasse Rempe-Gillen and describes it as a conjecture.