Piecewise-quasilinearity of stable commutator length in free products of cyclic groups

Let $G_j^k = \langle a_j \rangle / \langle a_j^{o_{j,k}} \rangle = \mathbb Z/o_{j,k}\mathbb Z$ and let $G_k = *_j G_j^k$. Fix $\Gamma$ such that $\Gamma \in B_1^H(G_k)$ for all $k$; this requires only that $\Gamma$ have the appropriate number of inverse pairs in the free factors. Piecewise-quasilinearity conjecture. For $o_{j,k}$ sufficiently large, the stable commutator length ${\textnormal{scl}}_{G_k}(\Gamma)$ is piecewise-quasilinear in the variables $1/o_{j,k}$. The conjecture predicts the eventual piecewise-quasilinear behavior of stable commutator length as the orders of the cyclic factors grow, extending the quasipolynomial phenomena observed in computations for free products of cyclic groups.