Lipschitz extension spectral synthesis conjecture for virtual persistence diagram groups

Let $(X,d,A)$ be a separable metric pair, let $E$ be a complex Banach space, and let $f\colon K(X,A)\to E$ be Lipschitz. Lipschitz extension spectral synthesis conjecture. Every such $f$ has uniformly discrete spectral synthesis as in the definition of uniformly discrete spectral synthesis. That is, Lipschitz continuity is conjectured to be sufficient, as well as necessary, for uniformly discrete spectral synthesis of these extensions.