Unique ergodicity under almost-everywhere relative deformation

Let $\mathbb{R}^d_+$ be the positive cone, and let $\mathrm{REL}\subset\mathbb{R}^d$ be the relative subspace. Let $\mathbf{a}\in\mathbb{R}^d_+$ be uniquely ergodic.

Relative unique-ergodicity conjecture. There is a neighborhood $\mathcal{U}$ of $0$ in $\mathrm{REL}$ such that $\mathbf{a}+\mathcal{U}\subset\mathbb{R}^d_+$ and $\mathbf{a}+\mathbf{b}$ is uniquely ergodic for almost every $\mathbf{b}\in\mathcal{U}$. This is proposed as a special case motivated by the failure of the paper's horocycle-flow methods to handle lines tangent to the relative subspace. Its status is open in the supplied source.