ANSWER: In brief, that depends on who the eternal security proponent is. Some have much more to give up by turning from their beloved and popular doctrine to the truth. Examples of such are pastors, evangelists, seminary teachers, district supervisors, ministry directors, and the like. People in that kind of position could be jeopardizing their livelihood, church position, pension, etc. by their change.
Hence, many of these will tend to close their eyes to the compelling arguments advanced by their opponents, magnify their own weak arguments, so-called proof texts, etc. to justify what they have been doing and teaching. Their rejection will allow them to continue as before with their short-lived life and job, as the rich, young ruler did after hearing and dismissing the truth from the Lord Jesus. Many want to believe in eternal security and will refuse to change, even after being confronted with the vital truth.
All other eternal security advocates who don't have so much to lose in this life (or perhaps sin to cling to) could be more apt to accept the evidence, as disturbing as it might be to them, but it will still take more - a heart for truth.
Some might try to counter this answer and say, "But wait, the ones with the church and ministry position are wiser than the others who are not in a leadership position." Such an answer is without Scriptural backing as the reason why some reject the truth and others don't. According to God's word, the ones who are wise and have understanding are the people who hear the word of God, put it into practice, fear God and shun evil (Mt. 7:24-27; Job 28:28; Prov. 1:7). It takes a heart to receive the truth (to do and believe the right things, as defined in Scripture) that results in blessed changes in a person's life, especially with eternity in mind. Counting the cost for all people who really want the truth first and foremost, will be a small price, because they desire to enter God's paradise kingdom more than anything this life can offer. May this book (The Myth Of Eternal Security) find its way into the possession of many such people for their own soul's sake.
Contact: Evangelical Outreach