General order convergence conjecture for subdivision schemes

Consider $d53d^n$ endowed with the metric $d(x,y)=\lVert x-y\rVert$, where the norm is any norm on $d53d^n$. Let $S_2$ be a subdivision scheme on $d53d^n$ satisfying, for $n=0,\ldots,m$,

$$\delta\left(\frac{\Delta^n(S^{L_n}(\mathcal{P}^{j}))}{2^{-n(j+L_n)}}\right)\leq \mu_n\,\delta\left(\frac{\Delta^n(\mathcal{P}^{j})}{2^{-jn}}\right),\qquad \mu_n\in(0,1),\quad L_n\in\mathbb{N}.$$

Let $S_1$ be a subdivision scheme on $d53d^n$ defined relative to the same parameter sequences as $S_2$. Suppose that $S_1^{L_m}$ is in proximity of the second type of order $m$ with $S_2^{L_m}$ in $d53d^n$. General order convergence conjecture. Then $S_1$ converges to a $C^m$ limit.

This conjecture extends the preceding first-order conclusion: convergence follows from the stated contractivity conditions and the order-$m$ proximity relation, while convergence of divided differences to the derivative is expected to yield the asserted differentiability of the limit curve. The general order case remains unproved in the supplied context.