It was previously asked if there exist Boolean circuits of treewidth $O(\log n)$ that compute the majority function $\text{MAJ}_n$ on $n$ inputs. While a construction using online algorithms and the related concept of online-width affirms this, in the original question the point was raised whether any of the known quasilinear sorting networks (which induce circuits for $\text{MAJ}_n$) share this property. As far as I'm aware, this was left unanswered.
Here, the treewidth of a sorting network is the treewidth of the underlying undirected graph that has as vertices the comparators, and each vertex has degree four (corresponding to the two inputs and outputs).
Hence, as a follow-up question:
Are there sorting networks of size $n\cdot \text{polylog}(n)$ and (poly)logarithmic treewidth?
More generally, I am interested in sorting networks of size $s$ and treewidth $\tau$ such that $s\tau^2 \leq n^{1.99}$. Conversely:
Are there lower bounds that rule out such networks?
One thing I could think of is that sorting networks should have high expansion, but that is vague.