It's not true. There should be zero closers drafted in the first round. Many good fantasy players don't even think about drafting a closer until round 7 or even later.
Typically, you should look to have at least 2 closers on your team by the end of the draft.
"And so he spoke, and so he spoke, that lord of Castamere. But now the rains weep o'er his hall, with no one there to hear." - The Rains of Castamere