Oil prices aren't high right now. In fact, they are unusually low. Gasoline prices would have to rise by another $0.65 to $0.75 per gallon from where they are now just to be "normal". And, because gasoline prices are low right now, it is very likely that they are going to go up more--perhaps a lot more.
Woodhill goes on to argue that the price of oil in terms of gold is low. In other words, he takes the price of oil in gold as the relevant price rather than the price of oil in dollars.
But let's say for a minute that Woodhill is right and that we should judge the price of oil in gold. Let's look at some recent history. On Feb. 3, the price of oil was $97.84 and on Feb. 24, just after he wrote, it had risen to $109.77. That's a 12% increase. Now if what's really matters is the price of oil in gold, then there shouldn't have been much of a change. So let's look at the price of gold over that same time. On Feb. 3 it was $1737.90. On Feb. 24, it was $1775.10. That's a 2% increase. That implies, using basic arithmetic, a 10% increase in the price of oil in terms of gold.
So the apparent constancy that Woodhill thinks should exist between gold and oil prices does not exist.
Furthermore, if what's really going on, as the headline says, is that the dollar is falling, then other prices besides gold prices should have risen in terms of the dollar. In other words, the inflation rate should be relatively high. The inflation rate for February is not due out until March 16. But I'm quite confident that it won't show even a 1% increase in a month (which would be 12% on an annual basis.)
An economics version of Occam's Razor applies here. When there are specific factors in a particular market that should be expected to affect prices in that market, look at those factors first before looking at other factors that should be expected to affect prices overall, especially when prices overall are not changing much.