Resonance between planar self-affine measures

We show that if $\{{\phi }_i{\}}_{i\in \mathrm{\Gamma }}$ and $\{{\psi }_j{\}}_{j\in \mathrm{\Lambda }}$ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures $\mu$ and $\nu$, the inequality $${\dim }_{\mathrm{H}}(\mu \ast \nu )<\min \{2,{\dim }_{\mathrm{H}}\mu +{\dim }_{\mathrm{H}}\nu \}$$ implies that there is algebraic resonance between the eigenvalues of the linear parts of ${\phi }_i$ and ${\psi }_j$. This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.