Perhaps an additional linkage part is needed. Again, see photos for specifics, I cannot find any engraved part numbers on it.
Lathe and collet drawbar shown in one photo are not included, just to show how it fits. The item "Collet closure assembly for South bend lathe 3c" is in sale since Sunday, November 7, 2021. This item is in the category "Business & Industrial\CNC, Metalworking & Manufacturing\Workholding & Toolholding\Workholding\Chucks\Collet Chucks". The seller is "427m6es" and is located in Spring, Texas. This item can be shipped to United States.