From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive convolution, free multiplicative convolution and free compression describe respectively the addition, multiplication, and taking minors of large random matrices. I have a question about an observed relationship between free additive convolution and free compression.
Free additive convolution. Let $N$ be large, and let $A = \mathrm{diag}(a_1,\ldots,a_N)$ and $B = \mathrm{diag}(b_1,\ldots,b_N)$ be diagonal matrices with real eigenvalues chosen in such a way that their empirical spectra approximate probability measures $\mu$ and $\nu$ in the sense that $$\frac{1}{N} \sum_{i=1}^N a_i \approx \mu$$ and $$\frac{1}{N} \sum_{i=1}^N b_i \approx \nu.$$
Let $U$ be Haar distributed on the unitary group. According to a result in free probability, the empirical spectrum of the random matrix $A+U^*BU$ approximates a probability measure $\mu \boxplus \nu$ on the real line. This measure is known as the additive free convolution of $\mu$ and $\nu$.
Free compression. On the other hand, with $A$ as above, set $j = \tau N$. Then the empirical spectrum of the $j \times j$ principal minor of $U^*AU$ approximates a probability measure $[\mu]_\tau$ on the real line. Call this measure the $\tau$-compression of $\mu$.
The relationship between free additive convolution and free compression. Let $k \geq 1$ be an integer. Set $\tau = 1/k$. Then it has been observed, for instance in this paper
Shlyakhtenko, Dimitri; Tao, Terence, Fractional free convolution powers, ZBL07642254. with a version freely available on arXiv here
that up to a rescaling, the $k$-fold additive free convolution $\mu^{\boxplus k}$ and the $\tau$-compression of $\mu$ are the same measure. (More precisely, one needs to pushforward the $\tau$-compression under the map $x \mapsto kx$ to get the same measure.)
This fact can be proved using formal calculations in free probability; see e.g. Section 2.5.4 on Free probability in Tao's book on Random Matrix Theory.
Question. Without appealing to any free probability, but just properties of the unitary group, is there a heuristic way to see that free compression and free additive convolution are the same up to rescaling? I have in mind a possible geometric heuristic, associating the Haar unitary matrix $U$ as a uniformly chosen orthonormal frame of $\mathbb{C}^N$.
Honing in on the case $k=2$: Is there a heuristic way of guessing that for large $N$, the empirical spectrum of $A+U^*AU$ is, up to scaling, approximately the same as the top-left corner of $N/2 \times N/2$ of $U^*AU$?
