Absence of low temperature anomaly on the melting curve of $^4$He

We have measured the melting pressure and pressure in the liquid at constant density of ultra-pure $^4$He (0.3 ppb of $^3$He impurities) with the accuracy of about 0.5 $\mu$bar in the temperature range from 10 to 320 mK. Our measurements show that the anomaly on the melting curve below 80 mK which we have recently observed is entirely due to an anomaly in the elastic modulus of Be-Cu from which our pressure gauge is made of. We thus conclude that the melting pressure of $^4$He follows the $T^4$ law due to phonons in the whole temperature range from 10 to 320 mK without any sign of a supersolid transition.