Real-analyticity conjecture for the hairs of the exponential map

Let $f:\mathbb{C}\to\mathbb{C}$ be the exponential map $f(z)=e^z$, and let its hairs denote the corresponding invariant curves in the escaping set. Real-analyticity conjecture for exponential hairs. The hairs of $f:z\mapsto e^z$ are real-analytic curves. The source presents this as a long-standing open problem, apparently related to the preceding escaping-set conjecture.