Yes, but that wasn't known at the time and that's not how it was originally dealt with the first time it happened. Prior to knowing the tattoo was moved it was all:
"Ah well, I guess you're innocent, let us take off that death countdown"
Therefore implying that the process can be nulified. However fast forward five seconds...
"Actually Little Stark, she's got it now."
"OH MY GOD WHAT HAVE YOU DONE."
Which gives the impression that the whole concept of it being able to be taken away is bullshit.