  In \cite{HSZ23}, with Gus Schrader and Eric Zaslow we developed a
skein-theoretic version of cluster theory, and made a conjecture on the
pentagon relation for the skein dilogarithm. Here we give a topological proof
of this conjecture. Combining \cite{MS21} and \cite{BCMN23}, we get a
surjection from the skein algebra $\mathrm{Sk}^+(T - D)$ to the positive part
of the elliptic Hall algebra $\mathcal{E}_{q, t}^+$. Hence our pentagon
relation generalizes the ones in \cite{Z23} and \cite{GM19}.
