R Eli Mansour explains here
During the times of the Beth Hamikdash, those who were unable to bring
their sacrifices on the day of Shavuot itself – which in Israel is
celebrated only on the sixth of Sivan - were allowed to do so during
the six days following Shavuot, through the twelfth of Sivan
As such some don't say Tahanun in the week after Shavuot. The Mishna Brura 131:36 for instance writes
There are places that omit Tachnun for six days after Shavuot, because
the sacrifices of Shavuos have seven days of Tashlumin - compensation.
Similarly the Alter Rebbe of Chabad writes in his siddur
One may not fast from the first of Sivan until after Shavuos. [This
applies until, and including, the 12th of Sivan]
while in the Shulchan HaRav, he permits fasting from the 9th.
It is very likely your Chabad rabbi refers to these rulings.