Positive-measure intersections of Bedford-McMullen carpets under similitudes

Let $K=K(n,m,\mathrm{\Gamma })$ be a Bedford-McMullen carpet with $n>m\ge 2$ and $1<\mathrm{\#}\mathrm{\Gamma }<nm$, and let $\mu$ be a self-affine measure on $K$ with strictly positive weights. We prove that if a similitude $f$ satisfies $\mu (K\cap f(K))>0$, then its orthogonal part is diagonal or anti-diagonal. This holds without any arithmetic restrictions on the defining bases. For isometries, we further prove that if $\mu (K\cap f(K))>0$ then $f(K)$ and $K$ differ only by a translation, thereby establishing a strengthened form of the 2010 conjecture of Elekes, Keleti and Máthé in this planar self-affine context.