"What are the most challenging objections . . . ? "
After over half a century of reading and studying and meditating on the holy scriptures (Genesis to Revelation, 66 books, TR/KJV/YLT/EGNT)(1) I do not find any objection to the inspiration of scripture to have any weight whatsoever. None of the arguments, to me, can be called 'challenging' in the slightest degree.
The scripture, itself, in its entirety, in its veracity, in its integrity and in its power (that is to say the power of the word of God when the Spirit of God utters it to one's own spirit) is of such force that no objection of any kind can stand for a moment in the mind or heart.
There is nothing else on earth which compares, to the slightest degree, to these holy scriptures. Some have attempted to mimic them and at the age of seventeen, purely as an academic exercise, I glanced at these attempts. I found them to have nothing whatsoever in common with that which they attempted to mimic, and I never referred to them again.
(1) TR - The Textus Receptus, the 'Received Text' namely Beza, Stephanus, Elzevir, and Scrivener which agree together save in but few places, none of which are doctrinally significant.
KJV - The Authorised King James Version
YLT - Young's Literal Translation
EGNT - The Englishman's Greek New Testament (Stephens Text) which contains an excellent literal, interlinear translation
These are the 'scriptures' which fill my mind, govern my mentality, and remain firmly in my soul, to the exclusion of all other ideology.
In over half a century, nobody has ever uttered an objection to these scriptures, nor have I ever read one, which had any power whatsoever to dissuade me from them. None of the arguments were to the least degree logical, practical, cogent, or appealing.