The affine-irreducibility conjecture for dimensions of self-similar attractors

Let $G$ be the relevant group of similarities of $\mathbb{R}^{d}$, and let $\Phi\subseteq G$ be an affinely irreducible iterated function system with attractor $X\subseteq\mathbb{R}^{d}$. Write $\operatorname{sdim}X$ for its similarity dimension. A linear subspace $V<\mathbb{R}^{d}$ is $D\Phi$-invariant if it is invariant under every orthogonal part $D\varphi_i$ of a map in $\Phi$; it is nontrivial when $0<\dim V<d$. Affine-irreducibility conjecture. At least one of the following holds:

$$\dim X=\min\{d,\operatorname{sdim}X\};$$

there are exact overlaps; or there exist a nontrivial $D\Phi$-invariant subspace $V$ and $x\in X$ such that

$$\dim(X\cap(V+x))=\dim V.$$

This conjecture modifies the one-dimensional exact-overlap principle to account for dimension loss accumulating on invariant subspaces. The paper proves a weak version, while the full assertion is presented as an open problem.