Maximal strict scaling conjecture for scalable frames

Let $F$ be a scalable frame with minimal scalings v_j_{j\in I}, and let $J\subset I$ be inclusion-minimal subject to

$$\bigcup_{j\in J}\operatorname{supp}(v_j)=\{1,\ldots,k\}.$$

Suppose that

$$\{v_j\}_{j\in J_1}\mathbin{\dot\cup}\cdots\mathbin{\dot\cup}\{v_j\}_{j\in J_a}$$

is the smallest orthogonal partition of v_j_{j\in J}. Maximal strict scaling conjecture. There exists a scaling $c$ such that

$$EC(cF)=\left\{\bigcup_{j\in J_i}\operatorname{supp}(v_j):i=1,\ldots,a\right\}.$$

The conjecture asserts that an orthogonal partition associated with a minimal collection of minimal scalings covering all coordinates can be realized as the factor structure of some scaling. Such a scaling would provide a factor poset containing all possible factor posets of strict scalings, potentially allowing frame constructions with multiple representations in selected directions for signal and image processing applications.