Outerplanar graph polygon-chain determination conjecture

Let $G$ be a bi-connected outerplanar graph. Let $G^*$ be its inner dual, and let $P_1\cup\cdots\cup P_t$ be the polygon chain decomposition of $G$ corresponding to the maximal path decomposition of $G^*$. Write $\tau(P_i)$ for the number of spanning trees of $P_i$, and let $\mu(G)$ denote the minimum number of generators of the sandpile group $S(G)$. Polygon-chain determination conjecture. The quantity $\mu(G)$ is determined by the numbers

$$\tau(P_1),\ldots,\tau(P_t).$$

For polygon flowers, the analogous assertion follows from the paper's results; the conjecture extends this phenomenon to all bi-connected outerplanar graphs, and its resolution is not given in the source.