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I came across this interesting question from almost 7 years ago:

What are the current breakthroughs of Geometric Complexity Theory?

My question is quite simple: Have there been any breakthroughs in the previous 7 years?

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    $\begingroup$ As a general principle, if there is already an MO question of this type, you can type the titles of the papers mentioned in the MO answers into Google Scholar to see if they have been cited recently. Any breakthrough paper will cite recent important work. In this case, I'm pretty sure that there hasn't been anything in the past 7 years that can be called a "breakthrough" (though the definition of that term is murky). $\endgroup$
    – Timothy Chow
    Commented Jun 4 at 11:49

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As explained by Scott Aaronson, in his 2017 overview of Geometric Complexity Theory, the core problem it tries to solve is to prove Valiant's conjecture VP$\neq$VNP that the permanent requires exponential-size arithmetic circuits. This problem could be a stepping stone to the harder problem of a proof of P$\neq$NP. A recent (2023) summary of the – limited – progress of the field is given by Greta Panova (section 6):

The study of Kronecker and plethysm coefficients has led to the 2017 disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem – the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation theoretic multiplicities in further detail, possibly asymptotically.

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