It depends. If they are of saving faith, which trusts/believes in the resurrected Son of God, the Lord Jesus Christ, and that as He will save them on His account, since He is Divine and paid the price for their forgiveness by His crucifixion on the cross, then all his past sins are forgiven in conversion, whether he remembered and named them or not.
The believer is now accepted in the Beloved” (Ephesians 1:6) on His account, with His faith being counted for righteousness, (Romans 4:5), and is in a covenant with His Lord, a relationship. Therefore this faith, if it is Biblically real, will result in obedience, “things that accompany salvation,” (Hebrews 6:9) including asking forgiveness for sins once one is convicted in his heart of them.
For although he is saved as a believer, the believer is still accountable to God with whom he is in a relationship with, and sin is displeasing to God, but who deals with the believer as a son. (Hebrews 12)
Therefore the believer will have a contrite, penitent heart over sins in general, and confess them when convicted in his conscience of them. (This does not mean one cannot subdue his conscience for a time, placing conviction in the background as David must have done until fingered by Nathan), but when enlightened and he is convicted of his guilty then he will confess and repent from the sin.
If not, then he is acting contrary to saving faith. denying the faith, as he would be if he impenitently consciously practiced sin, including omission, such as not providing for dependents of his family. (1 Timothy 5:8)
However, the key here is faith, as faith in the Lord Jesus is how one is made accepted by God, not on the basis of how good the person is, (Ephesian 2:8,9) but saving faith is only that will effects obedience by the Holy Spirit. “For as many as are led by the Spirit of God, they are the sons of God.” (Romans 8:14)
Therefore if unconfessed sin is the result of one hardening one’s heart against the conviction of the Holy Spirit, which can even include attributing the evidence behind the conviction to the devil, as did certain of the scribes in the days of Christ on the earth; and or continues impenitently in known sin, then that would be a denial of faith, and thus an “evil heart of unbelief, in departing from the living God,” (Hebrews 3:12) making Christ of no effect, to no profit, falling from grace, (Galatians 5:1–4) drawing back unto perdition, (Hebrews 10:25–39) forfeiting what faith had appropriated.
However, if the unconfessed sin is due to ignorance - which are much the reality in the life of believers, or or forgetfulness, then this does not impugn the faith of the believer, for he would live out his pious faith if he was conscious of these sins, and convicted of his guilt. And which faith renders the believer washed, and justified in the name of the Lord Jesus, and by the Spirit of our God. (1 Corinthians 6:11)
And wherever Scripture clearly speak of the next conscious reality for believers then it is with the Lord, (Lk. 23:43 [cf. 2Cor. 12:4; Rv. 2:7]; Phil 1:23; 2Cor. 5:8 [“we”]; 1Cor. 15:51ff’; 1Thess. 4:17)
And the next transformative experience that is manifestly taught is that of being like Christ in the resurrection. (1Jn. 3:2; Rm. 8:23; 1Co 15:53,54; 2Co. 2-4) At which time is the judgment seat of Christ, which is the only suffering after this life, which does not begin at death, but awaits the Lord’s return, (1 Corinthians 4:5; 2 Timothy. 4:1,8; Revelation 11:18; Matthew 25:31-46; 1 Peter 1:7; 5:4) and is the suffering of the loss of rewards (and the Lord’s displeasure) due to the manner of material one built the church with, which one is saved despite the loss of such, not because of. (1 Corinthians 3:8ff)