On the maximum twist width of delta-matroids

For a ribbon graph $G$, let $\gamma(G)$ denote its Euler genus. Recently, Chen, Gross and Tucker [J. Algebraic Combin. 63 (2026) 13] derived a formula for the maximum partial-dual Euler-genus $\partial\gamma_M(G)$ of a ribbon graph $G$. Their key finding is that $\partial\gamma_M(G)$ can be achieved by a partial dual with respect to the edge set of a spanning quasi-tree. Moreover, they proposed the following problem: Given a ribbon graph $G$, is there a sequence of edges $e_1,e_2,\dots, e_k$ such that $\gamma(G^{\{e_1, e_2,\dots, e_k\}})=\partial\gamma_M(G)$ and such that the sequence $$\gamma(G), \gamma(G^{\{e_1\}}), \dots, \gamma(G^ {\{e_1, e_2,\dots, e_k\}})$$ rises monotonically (i.e., never decreasing) to $\partial\gamma_M(G)$? Delta-matroids are set systems that satisfy the symmetric exchange axiom and serve as a matroidal abstraction of ribbon graphs. In this paper, we first show that the maximum twist width of a set system can be attained by twisting one of its feasible sets, which extends the result of Chen, Gross and Tucker to set systems. Then we solve the delta-matroid version of their problem, thereby providing an affirmative answer to the original problem for ribbon graphs.