Walker's piecewise quasilinearity conjecture for scl in free products of finite cyclic groups

Fix a rational chain $c$ in the free group $F_n$. For $\bm{o}=(o_1,o_2,\ldots,o_n)$ with $o_i\ge 2$, let $c_{\bm{o}}$ be the image of $c$ under the natural homomorphism

$$F_n\to *_i\mathbb{Z}/o_i\mathbb{Z}.$$

Here ${\rm scl}(c_{\bm{o}})$ denotes stable commutator length in the resulting free product, and a function is piecewise quasilinear in $1/o_i$ if there are some $p\in\mathbb{Z}_+$ and a finite partition of $\mathbb{Z}_{\ge 2}^n$ such that, on each piece and after fixing the congruence class of every $o_i$ modulo $p$, it is linear in the variables $1/o_i$.

Walker's conjecture. For any fixed chain $c$ in $F_n$, ${\rm scl}(c_{\bm{o}})$ is piecewise quasilinear in $1/o_i$: there are some $p\in\mathbb{Z}_+$ and a finite partition of $\mathbb{Z}_{\ge 2}^n$ such that on each piece, fixing any congruence class of each $o_i$ modulo $p$, ${\rm scl}(c_{\bm{o}})$ is linear in $1/o_i$.

The conjecture predicts the periodic behavior observed experimentally for stable commutator length in these families of free products; the source does not state a resolution, so its status remains open.