The core decomposition equality for polygonal presentations of compacta on surfaces

Let $K$ be a compactum in a closed surface $\mathcal{S}$, let $\mathcal{P}\subset\mathbb{C}$ be a polygon whose side identifications produce a quotient topologically equivalent to $\mathcal{S}$, and let $\tau:\mathcal{P}\to\mathcal{S}$ be the quotient projection. Put $L=\tau^{-1}(K)$. Let $\mathcal{D}_K^{PC}$ and $\mathcal{D}_L^{PC}$ be the core decompositions of $K$ and $L$, respectively, and let $\mathcal{D}_K^\tau$ be the finest upper semicontinuous decomposition of $K$ into subcontinua that split none of the images $\tau(\delta)$, for $\delta\in\mathcal{D}_L^{PC}$.

Core decomposition equality. The core decomposition of $K$ satisfies

$$\mathcal{D}_K^{PC}=\mathcal{D}_K^\tau.$$

This identifies the core decomposition on the surface with the finest decomposition compatible with the images of the polygonal core decomposition. The supplied text gives no resolution status beyond stating the equality in a conjecture environment.