No, he will remain in power, despite the rallies and discontent.
Despite the dissatisfaction of the generals with the Russian president for the appointment of a "non-military" Minister of Defense. He will not be overthrown until at least the end of 2024 for the simple reason that it is still better with a president than without him. Moreover, the president has already made concessions, signing an order to expand the staff and increase state support for the army.

But if the war drags on, then it is quite possible to expect an overthrow as a result of an uprising.
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The most likely scenario for concluding peace is an agreement mediated by third countries, if both sides consider it beneficial, but this option was postponed by the United States, ending negotiations for at least 6 months. The countries' goals are mutually exclusive, and neither side is yet ready to make concessions that the other deems acceptable.
Possible scenarios for the end of the war 1. Military exhaustion of one or both sides. 2. Political change is a change of government or an external course in one of the countries. 3. An agreement mediated by third countries, if both sides consider it beneficial. 4. Frozen conflict - cessation of active hostilities without final peace. 5. Victory in the war. Russia - if it starts mass mobilization and mass offensives. Ukraine - if the West starts fighting for it. Currently, the West (the United States, the EU, and NATO) supports Ukraine with weapons, intelligence, and finances, but it is not fighting for it.

Instead of starting negotiations, Ukraine was again pumped with weapons and money. They probably expect the Russian president to be overthrown.
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They do not think about the future of the country and the planet, they only think about their interests and enrichment
After the change of the president, the new administration begins a completely new policy that negates all the previous achievements of the old administration and the president. After that, no one will talk to them.

Their affairs are now worthy of some kind of street brawl, and not affairs of a national or planetary scale.
How does it happen, how much does it cost, and why does the duck limp? Questions about the transfer of power to the United States
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Four years old
Everything was done to make this war unstoppable. At least it will last until January 2029, when Trump leaves office, or earlier if he is assassinated.

It will be very difficult to stop this war with negotiations.
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