Contractibility and total semi-stability conditions of Euclidean quivers

We study the bounded derived category $\mathcal{D}$ of an Euclidean quiver, or equivalently, that of coherent sheaves on a tame weighted projective line. We give a description of the moduli space $\mathrm{ToSS}$ of the total semi-stability conditions on $\mathcal{D}$, which implies that $\mathrm{ToSS}$ can linearly contract to any chosen non-concentrated stability condition in it. For type $\widetilde{A_{p,q}}$, this gives an alternative proof of the contractibility of the whole space of stability conditions.