It seems unlikely.

We have no providence for the relic before it shows up and is purportedly discovered by St Helena. The story of its discovery in of itself contains fantastic elements and even within the story it was only identified because its proximity cured a noblewoman. There are good questions as to why the cross would have been buried at Golgotha in the first place and why it, the two thieves' crosses, and the nails used to crucify Jesus would have all been kept together and intact. There is also a question as to why a pagan temple would have been built on top of an execution site. And that's not the only story of the cross being recovered. The other origin stories seem equally fantastical.

What there doesn't seem to be any question about, is that by around 340 CE, there was a block of wood encased in a silver reliquary and venerated as a part of the true cross. How it got there and what it's true providence is, we simply don't know. There is a roughly 300 year gap when it disappears from the gospel narrative, and when it is "discovered."

However this relic was still in Jerusalem when the Sassanids sacked the city, and took it with them. This is what Heraclius is supposed to have recovered. But even that has been questioned. The simple fact is we don't have a proper chain of custody for the cross.