Homotopy exponents of mod2rmod2r Moore spaces

We prove that 2r+1⋅π∗(Pm(2r))=02r+1⋅π∗(Pm(2r))=0 provided m≥4m≥4 and r≥6r≥6. This is the best possible result. As well, for 2≤r≤52≤r≤5 we obtain upper bounds on the homotopy exponent of Pm(2r)Pm(2r).