Fibre stability for dominated self-affine sets

Let $K$ be a planar self-affine set. Assuming a weak domination condition on the matrix parts, we prove for all backward Furstenberg directions $V$ that $$\underset{E\in \mathrm{Tan}(K)}{\max }\underset{x\in {\pi }_{V^{\mathrm{\perp }}}(E)}{\max }\mathrm{}({\pi }_{V^{\mathrm{\perp }}}^{-1}(x)\cap E)=\mathrm{}K-\mathrm{}{\pi }_{V^{\mathrm{\perp }}}(K)\mathrm{.}$$ Here, $\mathrm{Tan}(K)$ denotes the space of weak tangents of $K$. Unlike previous work on this topic, we require no separation or irreducibility assumptions. However, if in addition the strong separation condition holds, then there exists a $V\in X_F$ so that $$\underset{x\in {\pi }_{V^{\mathrm{\perp }}}(K)}{\max }\mathrm{}({\pi }_{V^{\mathrm{\perp }}}^{-1}(x)\cap K)=\mathrm{}K-\mathrm{}{\pi }_{V^{\mathrm{\perp }}}(K)\mathrm{.}$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments.