Conjectured bounds for the simplification number of branched covering surface-knots

Let $(F,\Gamma)$ be a branched covering surface-knot of degree $N$. Let $u_w(F,\Gamma)$ and $u(F,\Gamma)$ denote the weak simplification number and simplification number, respectively. Define $c_{\mathrm{alg}}(\Gamma)$ as the sum of the absolute values of the numbers obtained, for $i<j$ and $i,j\in\{1,\ldots,N-1\}$, by subtracting the number of crossings of type $c_{j,i}$ from the number of crossings of type $c_{i,j}$. Conjectured simplification bounds. One should have

$$u(F,\Gamma)\leq u_w(F,\Gamma)+c_{\mathrm{alg}}(\Gamma),$$

and, successively,

$$u(F,\Gamma)\leq\max\{u_w(F,\Gamma),N-1\}+1,$$

and

$$u(F,\Gamma)\leq\max\{u_w(F,\Gamma),N-1\}.$$

The first inequality was previously proved only in a special case, while the latter bounds are proposed in connection with the proof of the preceding corollary. Their status remains open.