It is basically impossible for the Eastern and Western armies to meet. Mainly because the huge logistics is enough to bring down any empire. Emperor Wu of the Han Dynasty depleted the treasury and wealth of the two emperors Wen and Jing in a few battles against the Xiongnu, and had to carry out forced conquests on the people. If he were to carry out the labor expedition, he would soon be Just like Emperor Yang of the Sui Dynasty who conquered Goryeo, his country was destroyed and he died.
The Western Han Dynasty used troops in the Western Regions, with only ten or twenty thousand troops. Until the Qing Dynasty, the central army stationed in Xinjiang only had tens of thousands of troops. If you go further west, unless you are like the Mongolian army, there is no need at all. Logistics supplies can only be carried out by the army. If this kind of army encounters a powerful farming nation, it can only cause trouble, but it will never win. The Mongols are lucky (the Jin and Song Dynasties in the East have declined, the Arab Empire in the Middle East can only control Iraq for one generation, and there is no decent empire in Europe)
Zheng He's fleet can conquer the west, but it must Cross the Indian Ocean, reach the Cape of Good Hope in South Africa, turn to the Atlantic Ocean, and then cross the Atlantic Ocean from south to north before reaching Europe. How many people and supplies will be left in the fleet at that time? During the Opium War, Britain and France used India and Southeast Asia as their bases of attack. The East India Company had been operating in India for hundreds of years, and had colonies in South Africa and the west coast of Africa. It also sent only about 10,000 troops, which could only fight. But it cannot be conquered. Unless Zheng He established strongholds and colonies along the routes and then concentrated his efforts on attacking Europe, his fleet would not have much combat effectiveness.