Direct-summand conjecture for quasi-free Hilbert modules

Let $\mathcal{R}$ be a quasi-free Hilbert module of rank $k$, with $1 \leq k < \infty$, over $\mathcal{A}(\Omega)$. Let $\mathcal{R}_1$ and $\mathcal{R}_2$ be submodules such that

$$\mathcal{R}=\mathcal{R}_1\oplus_{\mathrm{alg}}\mathcal{R}_2.$$

Direct-summand conjecture. The modules $\mathcal{R}_1$ and $\mathcal{R}_2$ are quasi-free of ranks $k_1$ and $k_2$, respectively, and

$$k=k_1+k_2.$$

This proposed statement concerns the relation between free and projective Hilbert modules, analogous to the existence of holomorphic complements for some holomorphic subbundles of holomorphic bundles. The source presents it as a statement that should be true; no resolution is supplied here.