  Mim-width and sim-width are among the most powerful graph width parameters,
with sim-width more powerful than mim-width, which is in turn more powerful
than clique-width. While several $\mathsf{NP}$-hard graph problems become
tractable for graph classes whose mim-width is bounded and quickly computable,
no algorithmic applications of boundedness of sim-width are known. In [Kang et
al., A width parameter useful for chordal and co-comparability graphs,
Theoretical Computer Science, 704:1-17, 2017], it is asked whether
\textsc{Independent Set} and \textsc{$3$-Colouring} are $\mathsf{NP}$-complete
on graphs of sim-width at most $1$. We observe that, for each $k \in
\mathbb{N}$, \textsc{List $k$-Colouring} is polynomial-time solvable for graph
classes whose sim-width is bounded and quickly computable. Moreover, we show
that if the same holds for \textsc{Independent Set}, then \textsc{Independent
$\mathcal{H}$-Packing} is polynomial-time solvable for graph classes whose
sim-width is bounded and quickly computable. This problem is a common
generalisation of \textsc{Independent Set}, \textsc{Induced Matching},
\textsc{Dissociation Set} and \textsc{$k$-Separator}. We also make progress
toward classifying the mim-width of $(H_1,H_2)$-free graphs in the case $H_1$
is complete or edgeless. Our results solve some open problems in [Brettell et
al., Bounding the mim-width of hereditary graph classes, Journal of Graph
Theory, 99(1):117-151, 2022].
