Distinction of the Steinberg representation and the dual group of a symmetric space

We study the distinction of the Steinberg representation of a split reductive group $G$ with respect to a split symmetric subgroup $H\subset G$. We relate this distinction problem to a problem about the existence of a non-zero harmonic function on a certain hyper-graph related to $X=G\mathrm{/}H$. We verify the relative local Langlands conjecture for the Steinberg representation by showing that over a $p$-adic field the Steinberg representation is $H$-distinguished if and only if its Langlands parameter factors through the dual group of $X$.