The weak simplification number inequality for branched covering surface-knots

Let $(F,\Gamma)$ be a branched covering surface-knot. The quantities $u_w(F,\Gamma)$ and $u(F,\Gamma)$ denote, respectively, the weak simplification number and simplification number associated with $(F,\Gamma)$. Weak simplification inequality. One should have

$$u_w(F,\Gamma)\leq u(F,\Gamma).$$

This corrects an earlier claim whose proof was incomplete; the authors explain that the relevant 1-handles can be added from the outset. The conjecture asserts that weak simplification never requires more 1-handles than simplification.