The Great Wall of Persia

The bane of Persian empires has always been that while they accumulates immense wealth as they lie between the West and the East, people move, a lot, sometimes in huge numbers, armed, and on horseback. Could one of the Persian Empires have built a Great Wall in Central Asia to keep them out, in the style of the Great Wall of China? For some reason I think either the Seleucids or the Sassanians would be the best positioned to build it.
 
Perhaps they could extend the Great Wall of Gorgan from there to the Amu Darya (historically the border between Persia and Turan). But unlike China whose greatest threat by far lay to the north, Persia is surrounded by enemies. You'd need the western borders of Persia to be secured by vassal states like Armenia and no strong enemies like the Romans/Byzantines, and need something similar in the Arabian peninsula and along the Indus. Then a Persian ruler could perhaps decide to extend their fortifications to protect against the "Turanians".
 
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