The answer to this is generally yes but not always. Ultimately, the most important aspect is that your pace is improving at the same heart rate. Also, you should find that you can maintain that intensity longer as time goes by. One’s max HR changes with time and fitness, so the lactate / VO2 testing and approximation to race paces for intensity approximation is more valuable than simply HR.
None the less, yes, you should find yourself maintaining high HRs for longer periods of time. The goal is to increase your aerobic and anaerobic thresholds. This means that you can sustain higher percentages of your max heart rate – whatever number your personal max HR is.