Conjectural generalisation of the sewing lemma for approximate actions

Let $P$ be a metric space, and let $\big((M_x),\,(\mu_{xy})\big)$ be an approximate action of the pair groupoid $P\times P$, with each $M_x$ a complete metric space and $\mu_{xy}$ satisfying the estimates of Definition. Let $\mathcal G_P^{\smop{Lip-th}}$ denote the Lipschitz thin equivalence groupoid. Conjectural sewing lemma. There exists a unique action $\varphi$ of $\mathcal G_P^{\smop{Lip-th}}$ such that, for every representative path $\gamma:[0,1]\to P$ of a class $\mathbf g\in\mathcal G_P^{\smop{Lip-th}}$,

$$d_{\gamma(s)\gamma(t)}\big(\varphi_{[\gamma_{st}]},\,\mu_{\gamma(s)\gamma(t)}\big)\le C'|t-s|^{1+\varepsilon}.$$

Here $\gamma_{st}$ is the class of the restriction of $\gamma$ to $[s,t]$. For every choice of $\gamma\in\mathbf g$, $\varphi_{\mathbf g}$ is the limit in $C_{\gamma(1)\gamma(0)}$ of the compositions

$$\mu_{\gamma(0)\gamma(t_1)}\circ\mu_{\gamma(t_1)\gamma(t_2)}\circ\cdots\circ\mu_{\gamma(t_{k-1})\gamma(1)}$$

when the mesh of the subdivision $(0,t_1,\ldots,t_{k-1},1)$ tends to zero. One possible constant is

$$C'=2^{1+\varepsilon}g\big(\operatorname{Lip}(\gamma)|t-s|\big)\left(\sum_{i=1}^nC_i\right)\zeta(1+\varepsilon),$$

and the action satisfies

$$\operatorname{Lip}(\varphi_{\mathbf g})\le g\Big(\inf_{\gamma\in\mathbf g}\operatorname{Lip}(\gamma)\Big).$$

This is presented as a conjectural generalisation of the paper's main sewing theorem from local approximate flows to approximate actions over a metric parameter space. The statement asserts existence, uniqueness, convergence of subdivision compositions, a quantitative error estimate, and a Lipschitz bound, but the supplied text gives no resolution status.