The dimension of projections of planar diagonal self-affine measures

We show that if $\mu$ is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts, $${\dim }_{\mathrm{H}}\mu \circ {\pi }^{-1}=\min \{1,{\dim }_{\mathrm{H}}\mu \}$$ for any non-principal orthogonal projection $\pi$.