The representation claim for Heyting algebras with an E-pair

Let $\mathfrak{A}$ and $\mathfrak{B}$ be Heyting algebras such that $\mathfrak{A}\preccurlyeq\mathfrak{B}$. Let $a\in|\mathfrak{A}|$, and let $(a,a^{\ast})$ be an $\mathcal{E}$-pair in $\mathfrak{B}$. Write $\mathfrak{A}_{\tau_a}$ and $\mathfrak{B}_{\tau_a}$ for the corresponding $\tau_a$-expansions, and let $\mathfrak{A}_a$ denote the associated algebra; write $\delta[\mathfrak{A}_a]$ for its $\delta$-construction. Representation claim. If $\mathfrak{A}_{\tau_a}\vartriangleleft\mathfrak{B}_{\tau_a}$, then $\mathfrak{B}$ is isomorphic to $\delta[\mathfrak{A}_a]$. The claim is presented as the goal of the paper's argument; the supplied text does not establish whether it is resolved or remains open.