Almost-everywhere unique ergodicity along lines and analytic curves

Let $\mathbb{R}^d_+$ denote the positive cone, and let $\mathbf{a}\in\mathbb{R}^d_+$. A point is uniquely ergodic when the associated interval exchange transformation has a unique invariant probability measure.

The lines-and-curves conjecture. (Lines) If $\mathbf{a}$ is uniquely ergodic and $\ell$ is a line in $\mathbb{R}^d_+$ passing through $\mathbf{a}$, then there is a neighborhood $\mathcal{U}$ of $\mathbf{a}$ such that almost every $\mathbf{a}'\in\ell\cap\mathcal{U}$, with respect to Lebesgue measure on $\ell$, is uniquely ergodic. (Curves) If $\mathbf{a}(s)$ is an analytic curve in $\mathbb{R}^d_+$ whose image is not contained in a proper affine subspace, then for almost every $s$, with respect to Lebesgue measure on the real line, $\mathbf{a}(s)$ is uniquely ergodic. These are interval-exchange analogues of results for related Diophantine-approximation problems. The paper states that its horocycle-flow methods are insufficient for the line assertion, and the conjectures remain open in the supplied source.