The purpose of part number 6270 cable stop assembly is to prevent damage to the carburetor from excessive pilot force at the idle-cutoff position. If needed, install cable stop assembly on control wire(s). Locate stop so that when carburetor is at full idle, the cable stop is against part number 6271 cable assembly.
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Measuring the hole, it looks like this will accept a .078 diameter cable.
The id of the housing sets is .20". Part # 15-05393 is .215 so it may fit with some force.