I find it hard to believe there are relics of the True Cross. I don’t think the Romans were inclined to give away their execution devices for people to haul around and chop up. Wood was scarce in that area and had to be hauled from long distances. From what I have read of Roman executions, the vertical post of the cross was kept stationary while the crosspiece was carried by the prisoner and then used for the execution. To the Romans Jesus was just another criminal, nothing special to be done after the execution. When He was taken down, the wood was reused for other executions. Wood was not something to be wasted in that area of the world.
It does not bother me if others believe there is wood from the cross still around, but it does bother me that there is so much wood claiming to be the wood from the Cross floating around. I don’t like the idea that whatever relic I have cannot be absolutely positively guaranteed to be authentic, and after 2000 years and many, many people making claims, I don’t believe it can be proven.