The report attributed to their 12th Imam about al-Kafi was first mentioned by one of their scholars who lived during the Safavid era, however, there is no early source for the statement in any of their Hadith books so it is rejected by them as baseless.
However, some of their scholars including author of al-Kafi himself, as quoted by the brother above, claimed that the book is authentic or contains only authentic narrations.
Salaam alaykum wa rahmatullah,
The name "Al-Kafi" gives it away, doesn't it?
"You have mentioned that you are confused in the issues of the verification of hadith due to the difference in variously narrated texts and that you know the reason for variation but you do not find reliable people to discuss with. You have said that you wish you had a book sufficient (Kafi) that would contain all issues of the religion. A book that would provide a student all the material that he would need is urgently needed. A book is needed that would help people to have proper guidance in the matters of religion to follow the correct instructions of the truthful people (DivineSupremeCovenantBody) and the prevailing Sunnah, the basis of practices. So that one would fulfill his responsibilities towards Allah, the Most Majestic, the Most Gracious, and follow the Sunnah of (the Holy Prophet)." (page 27 of the document)
Source: https://www.holybooks.com/wp-content/uploads/Al-Kafi.pdf
Here is a better translation - this is what al-Kulayni writes in the introduction of his book:
وقلت إنك تحب أن يكون عندك كتاب كاف يجمع فيه من جميع فنون علم الدين ما يكتفي به المتعلم ويرجع إليه المسترشد ويأخذ منه من يريد علم الدين والعمل به
بالآثار الصحيحة عن الصادقين عليهم السلام والسنن القائمة التي عليها العمل وبها يؤدي فرض الله عز وجل وسنة نبيه صلى الله عليه وآله وقلت لو كان ذلك رجوت أن يكون ذلك سببا يتدارك الله تعالى بمعونته وتوفيقه إخواننا وأهل ملتنا ويقبل بهم إلى مراشدهم
“Verily, you solemnly wished that you possess
a book which is sufficient, brings together the entire Islamic sciences of the knowledge of religion within it, wholly satisfies the needs of the student, acts as a reference for the seekers of guidance, and would be used by those who want to attain the knowledge of religion and practice upon it by deriving
correct [şaĥīĥ] narrations of the truthful ones (as) and the upright and acted upon traditions from it—through which the compulsory duties of Allāh, the Powerful and Exalted, and the tradition of His Prophet (saws) can be fulfilled.
Their major scholar al-Hur al-Amili also believed the book is 100% authentic.
He states about the declaration of al-Kulaynī cited above:
وهو صريح أيضا في الشهادة بصحة أحاديث كتابه لوجوه منها قوله بالآثار الصحيحة ومعلوم أنه لم يذكر فيه قاعدة يميز بها الصحيح عن غيره لو كان فيه غير صحيح ولا كان اصطلاح المتأخرين موجودا في زمانه قطعا كما يأتي فعلم أن كل ما فيه صحيح باصطلاح القدماء بمعنى الثابت عن المعصوم بالقرائن القطعية أو التواتر
“This is also an explicit declaration of authenticity of the narrations in his book due to various points.
One of these points is: His statement: ‘Authentic narrations.’ It is well-known that neither did he mention a rule that distinguishes the rigorously authentic [şaĥīĥ] narration from the other categories in (the book), even if there is a non-şaĥīĥ narration in it, nor were the terminologies of the later scholars absolutely present during his time, as it will be further explained.
Thus, it is known that
all the narrations in it are correct [şaĥīĥ] by the terminology of the early scholars, with the meaning of being proven from the infallible on the basis of categorical indications or consecutiveness.”
(
Khātimat Tafşīl Wasā’il al-Shī`a ilá Taĥşīl Masā’il al-Sharī`a, of al-Ĥurr al-`Āmilī (d. 1104), volume 30, page 196 [Qum])
You can read more here:
https://web.archive.org/web/20120205035630/http://www.sunnidefense.com/exp/content/11-sh%C4%AB%C4%AB-scholars-and-authenticity-their-texts