The price of diesel has been high for a long time which doesn't cause inflation but inflation of prices. Also inflation in itself has caused diesel to rise as well so there are two reasons for the high prices - terrible energy policy and inflationary spending - both caused by the government (and Americans in general because we vote for more and more government spending). But to the consumer they don't care in a sense what is to blame for higher prices because either way it causes them serious pain.
What I liked most about your post is that you said it COULD bring prices down. Fuel surcharges, pallet charges and other fees added to deliveries very rarely go away easily. Only after a trucking outfit is getting beat badly by the competition or severe slowdown in shipments will cause the trucking prices to drop to generate some business.
I am basing the diesel connection to inflation based on a report from a diesel truck operator. In the report he spelled out changes in his diesel fuel cost last month compared to a 24 months back. From memory, it was close to triple high cost currently.
I am sure you are well aware from farms & mines to factories, from factories to warehouses, from warehouses to retail facilities, everything moves with diesel fuel.