None of the Imams and Mujtahideen have regarded it permissible to do masah on the type of wollen, cotton and nylon socks that are available today. You have understood it incorrectly that there is a difference of opinion regarding this matter. In fact the opinion of impermissibility is unanimous regarding such thin socks.

All fuqaha unanimously agree that masah on thin socks through which water can be strained, need to be tied to stay erect and cannot be used to walk a mile or two is not permissible and neither is masah on shoes permissible. And because the socks we have today are thin and they do not pass the 3 conditions stated, hence masah on them is not permissible under any condition. The person who does masah on them, then according to Imam Abu Hanifa, Imam Shafi'i, Imam Malik and Imam Ahmed, in fact according to any mujtahid, the wudhu of such a person will be invalid.

Sheikh ul hadith Abdur Raheem Abu Nauman briefly answers the popular question "Is masah over the socks allowed?" from Qur'an and Sunnah.