A negative answer to Kang–Jäger's Question 1.10 on permutative lifts

We answer Kang–Jäger's Question 1.10 (arXiv:2609.01140v1) negatively, including its strictly ergodic and Toeplitz versions. We construct one aperiodic regular Toeplitz subshift over the binary alphabet such that, for every permutative block code of every length ell >= 2, the entire inverse image is conjugate over the base to the product with the identity on 2^(ell-1) points. Each lift therefore has exactly 2^(ell-1) clopen strictly ergodic components. The construction uses recognizable equal-length words with factorial repetition counts: these annihilate every finite permutation action, while the proportion of columns on which the two words differ is exactly 3^(-n).