Let $(X_k,\omega_k)$ be the symplectic blow-up of the projective plane at $k$ balls, $1\leq k\leq 9$, of capacities $c_1,\ldots, c_k$. After reviewing some facts on Kahler cones and curve cones of tamed almost complex structures, we will give sufficient conditions on two sets of capacities $\{c_i\}$ and $\{c_i’\}$ for the associated symplectomorphism groups to be homotopy equivalent. In particular, we will explain when those groups are homotopy equivalent to stabilisers of points in $(X_{k-1},\omega_{k-1})$. We will discuss some corollaries for the spaces of symplectic balls.