Embedding cones over arc-smooth continua into their hyperspace of subcontinua

Given a metric continuum X  , let C(X)C(X) be the hyperspace of subcontinua of X   and Cone(X)Cone(X) the topological cone of X. We say that a continuum X   is ordered cone-embeddable in C(X)C(X) provided that there is an embedding h   from Cone(X)Cone(X) into C(X)C(X) such that, for each x in X  , h(x,0)={x}h(x,0)={x} and h(x,s)h(x,s) is properly contained in h(x,t)h(x,t) whenever s