Extension of dual subspaces invariant under an algebra

Phillips' known hypothesis concerning the extension of dual pairs of subspaces $\{\mathfrak L_1^0,\mathfrak L_2^0\}$, invariant under a commutative $J$-symmetric algebra $R$ in a Hilbert space $\mathfrak H$ , to a dual pair of maximal subspaces $\{\mathfrak L_1,\mathfrak L_2\}$, invariant under $R$ is established in the case where a dual pair of maximal subspaces exists $\{\mathfrak F_1,\mathfrak F_2\}$, invariant under $R$ with $\overline{\mathfrak F_1\oplus\mathfrak F_2}=\mathfrak H$, and the pair $\{\mathfrak L_1^0,\mathfrak L_2^1\}$ consists of $J$-neutral subspaces.