So my question is straightforward. Let $\mathfrak{X}$ be a (complex, if necessary) Banach space and $K\colon\mathfrak{X}\to\mathfrak{X}$ a nonzero compact operator. Denote by $\mathcal{C}(K)$ the commutant of $K$—i.e. the algebra of operators that commute with $K$, and $\mathcal{C}^2(K):=\mathcal{C}(\mathcal{C}(K))$.
For every $T\in \mathcal{C}^2(K)$ that is not a multiple of the identity, is it the case that $\mathcal{C}^2(T)$ also contains a nonzero compact operator?