Decomposition conjecture for normalized gauge-norm Hardy spaces

Let $\alpha$ be a continuous $\|\cdot\|_1$-dominating normalized gauge norm, and let $H^{\alpha}$ and $M_{\alpha}(z^n)$ be the associated Hardy space and invariant subspace. Decomposition conjecture. Is it true that

$$H^{\alpha}=M_{\alpha}(z^n)\oplus zM_{\alpha}(z^n)\oplus\cdots\oplus z^{n-1}M_{\alpha}(z^n)?$$

The decomposition is established earlier under the stronger assumption that $\alpha$ is continuous and rotationally symmetric, while the source leaves its validity for general continuous $\|\cdot\|_1$-dominating normalized gauge norms open.