The cointegration test you want to run is log of price (using real prices) and the log of price and log of time model derived from regression. I did the Johansen cointegration test with these time series and I found a cointegration equation with p=0.001. Ym1 is my matrix with 2 time series, log of prices and my power law model with power 6.

Results Summary (Test 1)

Data: Ym1
Effective sample size: 3454
Model: H1
Lags: 0
Statistic: trace
Significance level: 0.05

r h stat cValue pValue eigVal
— — — — — — — — — — — — — — — — — — — —
0 1 5169.5680 15.4948 0.0010 0.7759
1 0 3.7049 3.8415 0.0543 0.0011