On the similarity of powers of operators with flag structure

Let $\mathrm{L}^2_a(\mathbb{D})$ be the classical Bergman space and denote $M_h$ for the multiplication operator by a function $h$. Let $B$ be a finite Blaschke product with order $n$.An open question proposed by R. G. Douglas is whether the operators $M_B$ on $\mathrm{L}^2_a(\mathbb{D})$ similar to $\oplus_1^n M_z$ on $\oplus_1^n \mathrm{L}^2_a(\mathbb{D})$? The question was answered in the affirmative, not only for Bergman space but also for many other Hilbert spaces with reproducing kernels.Since the operator $M_z^*$ is in Cowen-Douglas class $B_1(\mathbb{D})$ in many cases, Douglas's question can be expressed as a version for operators in $B_1(\mathbb{D})$, and it is affirmative for many operators in $B_1(\mathbb{D})$.A natural question is how about Douglas's question in the version for operators in Cowen-Douglas class $B_n(\mathbb{D})$ ($n>1$)? In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class $B_2(\mathbb{D})$, and give a negative answer.This indicates that Douglas's question cannot be directly generalized to general Hilbert spaces with vector-valued analytical reproducing kernel.