This note gives a numerical investigation for the popular high resolution conservative schemes when applied to inviscid, compressible, perfect gas flows with an initial high density ratio as well as a high pressure ratio. The results show that they work very inefficiently and may give inaccurate numerical results even over a very fine mesh when applied to such a problem. Numerical tests show that increasing the order of accuracy of the numerical schemes does not help much in improving the numerical results. How to cure this difficulty is still open.