How is the $\chi$ step of the Keccak permutation invertible?

I would like to understand how Keccak's permutation function is reversible. The difficulty I have is with the $\chi$ step that uses the and operator which is not revertible. All the other suboperations are using xor and rotations which can be reversed.

In Keccak Reference, the authors said this:

$\chi$ is invertible but its inverse is of a different nature than χ itself. For example, it does not have algebraic degree 2. We refer to [20, Section 6.6.2] for an algorithm for computing the inverse of $\chi$.

Where [20] was:

[20] J. Daemen, Cipher and hash function design strategies based on linear and differential cryptanalysis, PhD thesis, K.U.Leuven, 1995.

It's quite hard to find it online, here's a link but I'm not sure if it'd suffer from link rot.

UPDATE

I know it's hard to understand it, and I myself had gave up. $\chi$ is the XOR of AND function on bits, and I tried to enumerate output of the AND operation to see if it's unique.

But that's expectable. $\chi$ is the "SBox" of the Keccak permutation (being non-linear). So it's totally normal for its mechanism to be arcane.

According to KeccakTools, the inverse of χ is given by the inverseChi function, i.e.:

template<class Lane>
void KeccakF::inverseChi(vector<Lane>& A) const
{
    for(unsigned int y=0; y<5; y++) {
        unsigned int length = 5;
        vector<Lane> C(length);
        for(unsigned int x=0; x<length; x++) C[index(x)] = A[index(x,y)];
        for(unsigned int x=0; x<(3*(length-1)/2); x++) {
            unsigned int X = (length-2)*x;
            A[index(X,y)] = C[index(X)] ^ (A[index(X+2,y)] & (~C[index(X+1)]));
        }
    }
}

There is a paper titled “The Inverse of $\chi$ and Its Applications to Rasta-like Ciphers” [F. Liu, S. Sarkar, W. Meier, T. Isobe]:

https://eprint.iacr.org/2022/399

In this work, we give a very simple formula of $\chi^{-1}_{n}$ that can be written down in only one line and we prove its correctness in a rigorous way.