Of course the Holy Spirit will remain on earth after the resurrection because God is an omnipresent Spirit, and there cannot be any place that He does not dwell, lol.
The Bible says the saints on earth cannot be gathered unto Christ until the man of sin claims to be God by harnessing the power of nature, and there is no personal God, which happens in the middle of the 7 years period.
That is because as long as the world can repent of their sins, and be saved, the saints are not going to be resurrected, and when the world follows the beast, and takes the mark which has to do with technology, then the saints on earth will be gathered unto Jesus.
But the Bible says the beast makes war against the saints, and prevails against them, and shall destroy the mighty and holy people, and they are given in to his hands for the last three and one half years.
There is only 2 resurrections, one of the saints before the millennial reign, and one after the millennial reign, and whoever does not get caught up in the first resurrection has to go to the second resurrection.
Also the Gospel must be preached in to all the world as a witness unto all nations, and then the end will come, which then comes the end when the Son shall deliver up the kingdom to the Father.
So the world knows about the Gospel, and Jesus said whoever does not get caught up in the resurrection claiming Christ shall have their portion with the hypocrites, there will be weeping and gnashing of teeth, and once He shuts the door and they knock it is too late.
So once the resurrection happens no Gentile can be saved, for if they were of the world they rejected the truth for all will know of the Gospel, and if they were claiming Christ and did not get resurrected they were hypocritical and lost the chance at salvation.
If the resurrection of the saints on earth happens pre-7 years period no Gentile can be saved after that point.
But the beast makes war against all saints worldwide, so there has to be Gentile saints on earth, which there cannot be if it is pre-7 years.
And it cannot be the Jews for they are protected from physical attack on Israel, and all the Jews are back on their land and not scattered in the world anymore because of the Middle East peace treaty, which God said He is not going to leave any of the Jews in the nations anymore so He can turn Israel to the truth.
Also the wrath of God is not in the 7 years period, for the first three and one half years salvation is still available to the world, and the 6 trumpets are warnings to the world to turn to God, for He is giving the world a chance to turn to Him and cease their agenda of trying to establish peace and believing in evolution and they can still evolve with no personal God above them.
And the beast has power to rule for the last three and one half years, so the wrath of God is not there for if it was how does the beast have power to rule.
The wrath of God is after the 7 years period.