Let $H$ be a complex infinite dimensional separable Hilbert space. There are various extensions of the following well known result:

Theorem(Lomonosov): Every nonscalar $T \in B(H)$ which commutes with a nonzero compact operator $K$ has a nontrivial hyperinvariant subspace.

It has been shown that there exist operators $T$ which do not commute with any nonzero compact $K$. This led to the following two generalisations; the first one can be found, for instance, in the book *Kubrusly, C. S. Hilbert space operators. Birkhauser, Boston, 2003* (Problem and Solution 12.4), while the second one is obtained in *Lauric, V. (1997). Operators $\alpha$-Commuting with a Compact Operator. Proceedings of the American Mathematical Society, 125(8), 2379-2384*.

Theorem: Let $T \in B(H)$ be nonscalar. If there exists a nonzero compact $K$ such that $\operatorname{rank} (TK-KT) \leq 1$, then $T$ has a nontrivial hyperinvariant subspace.

Theorem: Let $T \in B(H)$ be nonscalar. If there exists a nonzero compact $K$ such that $TK= \alpha KT$ for some $\alpha \in \mathbb{C}$, then $T$ has a nontrivial hyperinvariant subspace.

I was wondering if the following natural generalisation is true: if there exists a nonzero compact $K$ such that $\operatorname{rank}(TK - \alpha KT) \leq 1$ for some $\alpha \in \mathbb{C}$, then there exists a nontrivial hyperinvariant subspace.