Optimality conjecture for the local-set-based piecewise-constant wavelet basis

Let $\operatorname{W}$ be the local-set-based wavelet basis with the even partition. For a fixed sparsity level $K$, consider signals $\mathbf{x}$ satisfying

$$\left\|\Delta\mathbf{x}\right\|_0\leq K.$$

Optimality conjecture. The local-set-based wavelet basis $\operatorname{W}_{\rm LSPC}$ minimizes, among all orthonormal bases $\operatorname{F}$, the worst-case number of nonzero coefficients:

$$\operatorname{W}_{\rm LSPC}=\arg\min_{\operatorname{F}}\ \max_{\mathbf{x}:\,\left\|\Delta\mathbf{x}\right\|_0\leq K}\left\|\operatorname{F}^T\mathbf{x}\right\|_0,$$

subject to $\operatorname{F}$ being orthonormal. The conjecture asserts that this construction is optimal for promoting sparsity in piecewise-constant graph signals; the surrounding discussion establishes sparsity bounds for the proposed basis but provides no resolution of this global optimality claim.