This claim is grounded in tradition and not a modern fabrication. Though as Kilise mentions in the comments, the whole Quran had not been revealed when the verse 39:9 was revealed.
From Tafsir Ibn Kathir:
ابن عمر رضي الله عنهما يقرأ { أَمَّنْ هُوَ
قَانِتٌ ءَانَآءَ ٱلَّيْلِ سَـٰجِداً وَقَآئِماً يَحْذَرُ ٱلأَخِرَةَ
وَيَرْجُواْ رَحْمَةَ رَبِّهِ } قال ابن عمر ذاك عثمان بن عفان رضي الله
عنه. وإنما قال ابن عمر رضي الله عنهما ذلك لكثرة صلاة أمير المؤمنين
عثمان رضي الله عنه بالليل وقراءته، حتى إنه ربما قرأ القرآن في ركعة
Ibn Umar recited this verse and said that this virtue was present in
Uthman bin Affan, he used to offer Tahajud prayers at night in excess
and used to recite long portions of the Quran, so much that sometimes
he finished the Quran in a single Rakah.
From Tafsir Al Qurtubi
واختلف في تعيين القانت هاهنا، فذكر يحيـى بن سلام أنه رسول الله صلى
الله عليه وسلم. وقال ابن عباس في رواية الضحاك عنه: هو أبو بكر وعمر رضي
الله عنهما. وقال ابن عمر: هو عثمان رضي الله عنه. وقال مقاتل: إنه عمّار
بن ياسِر. الكلبي: صُهَيب وأبو ذرّ وابن مسعود. وعن الكلبي أيضاً مرسل
فيمن كان على هذه الحال
There is difference of opinion about who is referred to by qanit. Yahya bin
Salam said: This refers to the Messenger of Allah (ﷺ). Ibn Abbas said,
Zahak narrates from him, This refers to Abu Bakr and Umar, Ibn Umar
said: This refers to Uthman Bin Affan. Muqatil said: This refers to
Ammar bin Yasir. Kalbi said that this refers to Suhaib, Abu Zara and
Ibn Masud. It is also narrated from Kalbi that this means every person
who is in this state.