Frames for Sobolev spaces of differential forms

Let $\mathcal{M}$ be the compact Riemannian manifold considered in the paper, let $m=\dim\mathcal{M}$, and let $J$ be the integer from the frame theorem for $L^2$ spaces. For $k\in\{1,\ldots,m\}$ and $p\geq 0$, define

$$B^J_{k,p}=\{b^{i j_1\cdots j_k}_p:i\in\{0,1,\ldots\},\ j_1,\ldots,j_k\in\{1,\ldots,J\}\},$$

and

$$\widetilde B_{k,p}=\{\widetilde b^{i j_1\cdots j_k}_1:i\in\{0,1,\ldots\},\ j_1,\ldots,j_k\in\{1,2,\ldots\}\}.$$

Frames conjecture. The sets $B^J_{k,p}$ and $\widetilde B_{k,p}$ are frames for $H^1_k$, for every $k\in\{1,\ldots,m\}$ and $p\geq 0$.

The claim extends the established frame constructions for $L^2$ spaces of vector fields and forms and for order-one Sobolev spaces of $1$-forms. The source notes that the general construction has not been explicitly verified and suggests proving it by a similar argument or inductively in the Sobolev order; the conjecture therefore remains open.