The third order interval Gargantini method for the simultaneous inclusion of polynomial zeros was improved to the fourth order method by Carstensen and Petkovi\'c [An improvement of Gargantini's simultaneous inclusion method for polynomial roots by Schroeder's correction, Appl. Numer. Math. 13 (1994), 453–468]. We investigate the numerical stability of this improved method in the presence of rounding errors. The dependence of the convergence rate of the considered method on the magnitude of rounding errors is studied.