Nilpotency conjecture for the Collatz adjacency submatrices

Let $f:\mathbb{N}\longrightarrow\mathbb{N}$ be the shortcut Collatz function, and let $A$ be the infinite matrix with entries

$$a_{ij}=\delta_{f(i),j}.$$

For $n\geqslant 3$, let $C_n$ be the $(n-2)$-by-$(n-2)$ submatrix indexed by $\{3,\ldots,n\}$, with entries $c_{ij}=a_{ij}$. A matrix is nilpotent if some positive power of it is zero.

Nilpotency conjecture for the Collatz adjacency submatrices. If $n\geqslant 3$, then the matrix $C_n$ is nilpotent.

The paper presents this matrix statement as a matricial formulation equivalent to the aperiodic Collatz conjecture. Since the underlying Collatz assertion remains unresolved, this claim is open.