Purgatory, according to Catholic teaching, is a place or state after death where sinners go to be purged (cleansed) of their sins so they will be purified sufficiently to enter Heaven (Wikipedia). Protestants (and your author - until recently) have largely considered this teaching to be false.
It is the purpose of this study to embrace the parts that have scriptural support and reject the parts that we find to be false.
We will discover that the lake of fire is a state of purgatory.
"Purgatory" comes from latin purgatorium, from Latin purgare, a verb meaning to purge (or purify as related to sin)
Without extensively describing the Catholic view, I will mention in passing some beliefs, true and false.
Catholic doctrine is that if a person is good enough when they die, they go to Heaven. If bad enough they go to hell. Everyone in between goes to purgatory, where they stay until they have been purified sufficiently to go to Heaven.
In contrast, the Bible teaches that when a person dies they go into the grave until the resurrection (Ecclesiastes 9:5, Luke 14:14, John 5:29).
In this regard, there is no purgatory, but our disagreement is more related to "when" and "where" than "if".
On the other hand, if we closely study the lake of fire and accept that it is a figurative (symbolic) cleansing fire, NOT a literal fire, (See Lake of Fire) we find a cleansing, purifying purpose that is purging (purgatory). See Purpose of Yahweh's Fire
The information in the links above and below won't be duplicated here, so please study them. It is my hope that your study will find that:
1. The Lake of Fire is figurative, not literal (Hebrews 12:29).
2. It's purpose is for purification (cleansing, purging, so is truly purgatory).
3. People (sinners) are "thrown into" this lake of fire after the millennium (1000 years) and after the return of the New Jerusalem to the Earth and after their resurrection (Revelation 20:15).
4. This figurative fire will consume all sin.
5. The gates of the Holy City will never be closed. They are wide open, waiting for purged, repentant sinners to enter (Isaiah 60:11, Revelation 21:25).
6. ALL (you, me and EVERYONE) have sinned and are in need of the Saviour's fire (Mark 9:49, 1 Corinthians 3:13-15).
7. Each person is saved in their own turn and time (1 Corinthians 15:22-23, Philippians 2:10-11, 1 Timothy 4:9-11, Harvests of Redemption).
8. The Barley harvest will take the barley (righteous) to Heaven (see Barley Harvest).
9. The Wheat (see Wheat Harvest) and Grapes (see Grape Harvest) never go to Heaven but will be resurrected and harvested on the Earth after the millennium. Not going to Heaven does NOT mean they won't be saved (in their own turn) after being purged (cleansed).
It would be preferable to be part of the barley harvest and spend the millennium in Heaven, thereby avoiding the lake of fire.
Our Father is more loving and powerful than most people realize. He is willing and able to save even the wheat and grapes. Nothing will be destroyed (Isaiah 65:8). Nothing will be wasted (John 6:12). These harvests will yield an abundance, ready to enter the open gates of the Holy City after being purged in His Holy Fire, purgatory, the Lake of Fire.