This is probably true in this case. In the case of physical goods it is not always true. If there is continued demand but less of it, prices can rise for some goods because the fixed costs of manufacturing increase on a per item basis as you can't buy materials in as much volume. Increased costs of manufacturing an item will lead to higher prices for the consumer.

Thus reduced demand does not always lead to reduced prices.

However, even in your example, the higher prices will only come in time. This is assuming we are talking about a commodity good that is traded on an open market.

The immediate effect of a sudden reduction in demand is an increase in inventories on the sell side. If there is no expectation of resumption of demand, the inventory holders are likely to panic and sell at a reduced price to cut inventories. This is a classic case of "first out the door wins" and is what crashes are made of. They will also cut manufacturing (or mining or farming), but the affect of this on the price will only manifest once the reduction in production makes it through the supply chain. For something like corn, this could take upwards of a year.

Obviously, all this is affected by a large number of variables: warehousing costs, profit margins, manufacturing complexity and lead time, pricing power, tooling costs and other capital expenditures, etc.