Decomposition conjecture for finite Blaschke factors in rotationally symmetric Hardy spaces

Let $H^{\alpha}$ be the Hardy space associated with a continuous rotationally symmetric norm $\alpha$, let $B$ be a finite Blaschke factor of degree $n$, and let $e_{00},e_{10},\ldots,e_{n-1,0}$ denote the corresponding functions used in the decomposition. Decomposition conjecture. Is it true that

$$H^{\alpha}=e_{00}M_{\alpha}(B)\oplus e_{10}M_{\alpha}(B)\oplus\cdots\oplus e_{n-1,0}M_{\alpha}(B)?$$

The preceding decomposition is proved in the special case $B(z)=z^n$; its validity for general finite Blaschke factors is left unresolved in the source.