Infinite-dimensional operator invariant-subspace conjecture

Let $f_1,\ldots,f_l,g\in\mathbb{C}\langle\underline{x}\rangle$. For a complex Hilbert space $\mathcal{H}$, let $\mathcal{B}(\mathcal{H})$ denote the bounded operators on $\mathcal{H}$, and let $\underline{A}=(A_1,\ldots,A_d)\in\mathcal{B}(\mathcal{H})^d$. Infinite-dimensional operator conjecture. The following are equivalent: (i) for every complex Hilbert space $\mathcal{H}$ and every tuple $\underline{A}\in\mathcal{B}(\mathcal{H})^d$, every joint invariant subspace of $f_1(\underline{A}),\ldots,f_l(\underline{A})$ is invariant under $g(\underline{A})$; (ii) $g$ belongs to the unital $\mathbb{C}$-algebra generated by $f_1,\ldots,f_l$. The finite-dimensional version does not hold in the general non-homogeneous case for $l\geq2$, so this conjecture asks whether allowing evaluations on infinite-dimensional bounded operators restores the equivalence.