The isomorphism conjecture for notational and ordinal normal forms

Let $\mathcal{C} = C_0(\mathcal{J}) \cap \Omega$. Let $+_{\mathrm{NNF}}$ ("notational normal form") be the restriction of $+$ to pairs $(s,t) \in \mathcal{OT}$ such that $s+t \in \mathcal{OT}$, and let $+_{\mathrm{ONF}}$ ("ordinal normal form") be the restriction of $+$ to pairs $(\alpha,\beta) \in \mathcal{C}$ such that $\alpha+\beta$ is in Cantor normal form.

Isomorphism conjecture. There is an isomorphism

$$o:(\mathcal{OT},\leq,+_{\mathrm{NNF}})\longrightarrow(\mathcal{C},\leq,+_{\mathrm{ONF}}).$$

This asserts that the notation system equipped with its order and restricted addition is isomorphic to the corresponding ordinal structure in Cantor normal form. The supplied text gives no information about whether the assertion has been proved or remains open.