Dimension of diagonal self-affine measures with exponentially separated projections

Let $\mu$ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $\Phi = \{ (x_{1}, \ldots, x_{d}) \mapsto ( r_{i, 1}x_{1} + t_{i,1}, \ldots, r_{i,d}x_{d} + t_{i,d}) \}_{i\in\Lambda}$ on $\mathbb{R}^{d}$ and a probability vector $p = (p_{i})_{i\in\Lambda}$. For $1 \leq j \leq d$, denote the $j$-th the Lyapunov exponent by $\chi_{j} := \sum_{i\in\Lambda} - p_{i} \log | r_{i,j} |$, and define the IFS induced by $\Phi$ on the $j$-th coordinate as $\Phi_{j} := \{ x \mapsto r_{i,j}x + t_{i,j}\}_{i\in\Lambda}$. We prove that if $\chi_{j_{1}} \neq \chi_{j_{2}}$ for $1 \leq j_{1} < j_{2} \leq d$, and $\Phi_{j}$ is exponentially separated for $1 \leq j \leq d$, then the dimension of $\mu$ is the minimum of $d$ and its Lyapunov dimension. This confirms a conjecture of Rapaport by removing the additional assumption that the linear parts of the maps in $\Phi$ are contained in a 1-dimensional subgroup. One of the main ingredients of the proof involves disintegrating $\mu$ into random measures with convolution structure. In the course of the proof, we establish new results on dimension and entropy increase for these random measures.