Polynomial-time conjecture for subgraphs with boundedly many components

Let $G$ be a finite graph and let $H$ be a subgraph of $G$ with at most $k$ components, where $k$ is a natural number. The notions $S$-Eulerian and semi $S$-Eulerian mean, respectively, that $H$ has an $S$-Eulerian circuit or an $S$-Eulerian trail: a closed or open trail in $G$ containing every edge of $H$. Polynomial-time conjecture. For every natural number $k$, determining whether $H$ is $S$-Eulerian or semi $S$-Eulerian can be done in polynomial time. The paper establishes NP-completeness for unrestricted subgraphs and a linear-time algorithm for connected subgraphs; the conjecture proposes polynomial-time solvability for every fixed bound on the number of components.