We investigate experimentally the possibility of enhancing the production of $2^3S$ positronium atoms by driving the $1^3S$-$3^3P$ and $3^3P$-$2^3S$ transitions, overcoming the natural branching ratio limitation of spontaneous decay from $3^3P$ to $2^3S$. The decay of $3^3P$ positronium atoms towards the $2^3S$ level has been effciently stimulated by a 1312.2nm broadband IR laser pulse. The dependence of the stimulating transition efficiency on the intensity of the IR pulse has been measured to find the optimal enhancement conditions. A maximum relative increase of $ times (3.1 pm 1.0) $ in the $2^3S$ production efficiency, with respect to the case where only spontaneous decay is present, was obtained.