If we look at some short time periods, it is not unusual to observe VIX prices to look like a mirror image of SPX prices, just like in the chart on FT website. This pattern however is not robust - if we look at a scatterplot of SPX prices vs VIX prices over the last 20 years, it is clear that there is no correlation. It is also intuitive - S&P moves around all over the place, while VIX stays most of the time in a range.
Where correlation is robust is in returns - percent changes (daily, weekly, etc.) The scatterplot of daily returns for the last 20 years clearly shows strong negative correlation (-0.69) in the data.
The same pattern exists for weekly and monthly returns (I created the plots using non-overlapping returns)
From what I see in the data any correlation between SPX and VIX price levels is mostly spurious and coincidental.
What is equally puzzling is the authors' claim that correlation turned positive since October 22nd of this year. There's been only 8 trading days, and hardly any statistic should be calculated on only 8 observations, especially such a "noisy" one as correlation. Even if I were to agree with Ms Brooks methodology, the correlation observed is 0.35 +/- 0.37 (MSE using 10K bootstrap simulations) i.e. the real correlation cannot be reliably estimated based on such a short sample. The last plot is the histogram of correlation coefficient.