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Because Our Lord gave Satan 100 years do make his attempt to destroy the Church [Leo XIII]. The 100 years started in 1929 [Jesus to Sr. Lucia at Tuy].

... all speculation.  So, there are varying accounts of this story, one with 75 years, one with 100.  While Father Cekada claimed the story was fabricated, I actually tracked down there there was an original source that was very close to the story Father claimed was completely made up, but the missing element in the original was the specification of the timeframe, 75 or 100 years.  The original source of Pope Leo's vision was a well-respected secretary of Pope Leo XIII, whose account was then related by a highly respected Cardinal (Segura) who said he heard it directly from that secretary.  After Mass, as the story goes, Pope Leo did in fact have a vision where he froze and went pale, and after a long time, where people wondered if he had died, he immediately rushed out to his office and composed both the shorter-form St. Michael prayers and the longer Excorcism prayer.  Pope Leo told this secretary that he had seen a vision of the forces of Hell swarming over top of the Vatican with the intent to invade the Church, and some details can be inferred from the Exorcism prayers, about replacing the See of Truth with their throne of inquity so that when the Shepherd had been struck, the sheep would be scattered.

Now, Father Cekada pounces on this as well, claiming that this Throne "in the Holy Place" was a reference to the Quirinal Palace, a Castel Gondolfo like place that the Pope had controlled as an alternative residence, but had been taken over by the Italians, claiming that the reference in the prayers was to past events, including the Quirinal Palace as "the Holy Place".  Problem is that there's no evidence of Quirinal Palace EVER having been referred to as a Holy Place, as it was merely an administrative location when the Pope wasn't in Rome.

But, anyway, one version of the story say 75 years, and that could have begun in 1958/9 with the election of Roncalli.  How did Satan largely take control over the Church in 1929?  Yes, the pontificates of Pius XI and Pius XII were not particularly good for the Church, but I would say that it would be an exaggeration to say that one could mark some period of greater control starting from 1929.  NOW ... you definitely could surmise that this could have begun in 1958.

Lateran Treaty...1929....fwiw.

Yes, and I'm sure it factors in.  February 11, Lateran Treaty signed.  June 13, Our Lady requests consecration of Russia.

I know, of course, that the Dimonds Brothers trace the 10 kings to the 10 popes begining from that point forward as corresponding to the ones in the Apocalypse, and even the AntiPopes are included because a "priestly kingdom" is described, and through Ratzinger you had valid priests.