You are talking about the believer making a choice then. That in a time of increased lawlessness the believer is low on ability to endure, so they give up before the end.. therefore he does not reach that moment of being saved.
Has it been established yet what "the end" is.. and what "being saved" is?
It seems that you might be assuming what those two are referring to because you've stated that those whose agape grows cold go to hell. I don't see that being said in those verses, so I think that some clarification is in order?
The study of Mat.24:12-13 is to prove against OSAS which everyone I think understands "S" to mean 'saved' : as in born again, a citizen of heaven, a child of God. I want to state that I am fully aware of

.6:4-6 that states that it is possible to fall away from faith.. therefore OSAS is only contingent on the believer persevering in faith. It's not an automatic position.
However, a babe in Christ can loose their capacity to love if the pressure is great enough. That would not mean that the Father would assign them to hell. Now Mat.24:12-13 however seems to refer to adults. However, even adults can loose their love if sufficiently pressured also. What can be said about the verses are that most peoples love will grow cold.
While others who are rooted in God's love will withstand any pressure.
It seems to directly imply a love for the brethren which grows cold. It may imply indirectly that the love for God also grows cold. The apostle John wrote extensively in his epistle that one cannot separate the love of the brethren from the love of God. Yet he addressed the attitude among the believers that they thought that they could love God yet not love the brethren.
Did he say then that they were assigned to hell?