Mortad's conjecture on trivial domains of powers and adjoints

Let $n\in\mathbb{N}$. For a closed, densely defined operator $T$, write $D(T^k)$ for the domain of its $k$-th power and $D[T^{*k}]$ for the domain of the $k$-th power of its adjoint. Mortad's conjecture. For each $n\in\mathbb{N}$, there is a closed and densely defined operator $T$ such that $D(T^{n-1})\neq\{0\}$ and $D[T^{*(n-1)}]\neq\{0\}$, yet

$$D(T^n)=D(T^{*n})=\{0\}.$$

This conjecture seeks operators whose domains of powers become trivial at the $n$-th power while the preceding powers of both the operator and its adjoint have nontrivial domains. It is motivated by earlier counterexamples for squares and higher powers of closed, densely defined operators; its resolution is not supplied in the source.