Never heard of it. However, I occasionally tag along with a friend who goes to a Dutch-Portuguese Shul in Jerusalem and as I recall, the custom there is to only say the first paragraph of Aleinu.
The rabbi of the Spanish-Portuguese synagogue in London, R. Shem Tob Gaguine, writes in his Keter Shem Tob (vol. 1 p. 103) that such is the custom in London-Amsterdam (SPS). He suggests that perhaps they omit that section not to burden the congregation.
Rabbi Dr. David de Sola Pool notes (his Prayer Book (p. 104-5) that “some” add the section “ועל כן”, further indicating that it was not the Spanish-Portuguese custom.
From Wikipedia (this might help explain the different customs in general):
In the Middle Ages the custom grew up of reciting the first paragraph
every day, at the end either of the morning service alone or of all
the prayer services for the day. In the 16th century the kabbalist
Hayim Vital, recording the opinions of Isaac Luria, ruled that both
paragraphs should be included in all services, and should end with the
verse "on that day the Lord shall be one and His Name one". This has
been accepted in almost all communities except for the Spanish and
Portuguese Jews, who retain the "short Alenu". The custom
according to some North African prayer books is to recite the second
paragraph only at the conclusion of weekday morning services.