No, that's not what the evidence shows, eg: https://www.sciencedirect.com/science/article/abs/pii/S00472...

> Gruber is able to identify incidence on gross earnings as well as on employment by exploiting variation in payroll tax changes between firms. The benefit of the payroll tax cut is found to have been fully shifted to workers through higher earnings, with no significant employment effects. With similar objectives, Anderson and Meyer, 1997, Anderson and Meyer, 1998 use US firm-level micro data to measure the effects of changes in an experience rated Unemployment Insurance system. Payment variation between firms, due to the number of workers laid off subsequently claiming UI benefits, allows identification of the incidence of the tax on earnings. At the four-digit industry level, Anderson and Meyer find full shifting of the burden of higher payroll tax from employers to workers in the form of lower earnings. They report insignificant employment effects. We find strong evidence of partial shifting of the burden of income tax from worker to employer. Although income tax is incident on equilibrium wages, the tax burden is not fully shifted.