You made a logical mistake in cointegrating log of price with log of time that is meaningless. What you want to cointegrate is log of price and the power law in time model. That is the only meaningul comparison.
When you do that, hold and behold you find cointegration, using Johansen cointegration test, with p=0.004.
Also log of price passed an Augumented Dickey Fuller test when the model used is Autocorrelation plus Drift that will produce a “straight line” in a log log graph of price and time. Intuition and Understanding before Stats. Always.