Doubtful. Under basic economic theory, as demand goes down, so does price. The petroleum production that shuts down first will tend to be the most expensive production, causing a drop in the average cost, and price (we should see a similar effect in upstream products like crude).
The main way this effect breaks down is if the market shrinks so much that we loose the economy of scale; but it seams unlikely demand will fall far enough for that to occur.
That was not his point. When you extract oil, you don't really get a single product but multiple ones (heavy oils, lubricants, etc.)
Before you might be selling lubricants cheaply, because after refinement you had lots of lubricants as a by-product. If you need less car fuel, this changes the cost equation and lubricants (and other byproducts) might become more expensive even if car fuel becomes cheaper.
The main way this effect breaks down is if the market shrinks so much that we loose the economy of scale; but it seams unlikely demand will fall far enough for that to occur.