No, this is not true. Absolution is not dependent upon completing one’s penance. That said, if one goes into Confession with the intention of not doing the assigned penance, absolution would be invalid. Also, if one willfully chooses not to do his penance which was given for mortal sins confessed, it is considered grave matter. If it was only for venial sins, it is a venial sin to willfully neglect to do one’s penance.
With regards to this particular penance, I would either go back to the original confessor, or a different confessor and ask for a new penance that can be fulfilled right away while still doing what was originally requested of you for the spiritual benefits of it.