Low energy extension conjecture for matrix-valued positive definite functions

Let $\mathbb{F}$ be the free group, let $\mathbb{B}_r$ be the radius-$r$ ball, and let $\mathrm{NSPD}(r,d)$ denote the class of normalized strictly positive definite $d\times d$ matrix-valued functions on $\mathbb{B}_r$. For strictly positive definite functions on all of $\mathbb{F}$, let $\mathfrak{e}$ denote the relative energy defined through the associated transport operators. Low energy extension conjecture. For $r,d\in\mathbb{N}$, $\omega>0$, and $\mathsf{C}_1,\ldots,\mathsf{C}_n\in\mathrm{NSPD}(r,d)$, for every $m\in[n]$ there exists a strictly positive definite function $\widehat{\mathsf{C}}_m:\mathbb{F}\to\mathrm{Mat}_{d\times d}(\mathbb{C})$ such that

$$\mathsf{C}_m=\widehat{\mathsf{C}}_m\upharpoonright\mathbb{B}_r$$

and

$$\mathfrak{e}(\mathsf{C}_m,\mathsf{C}_k)=\mathfrak{e}(\widehat{\mathsf{C}}_m,\widehat{\mathsf{C}}_k)$$

for all $m,k\in[n]$. The supplied context identifies this as the paper's main harmonic-analysis conjecture, but does not explain the role of $\omega$ or provide evidence of resolution.