Almost slice-fullness conjecture for finite-dimensional left-Hilbert modules

Let $\mathcal{H}$ be a left-Hilbert module that is finite-dimensional as a vector space. Let $X$ be a measure space, let $\mathbb{F}$ be the underlying scalar field, and let $\mathcal{F}_{(X,\mu),\mathcal{H}}$ denote the space of continuous frames in $\mathmathcal{H}$ indexed by $X$. For $l\in\mathbb{N}^*$, consider continuous Bessel families $U(i)=(U(i)_x)_{x\in X}$ and frames $A(i)\in\mathcal{F}_{(X,\mu),\mathcal{H}}$ for $i\in[1,l]$. The finite-dimensional left-Hilbert-module conjecture. There exist $l$ manifolds $(D_i)_{i=1}^l$ in $\mathbb{F}^l$, each of codimension at least $1$, such that

$$\left(\sum_{i=1}^l c_i(A(i)-U(i))\in\bigcap_{i=1}^l\left(\mathcal{F}_{(X,\mu),\mathcal{H}}-U(i)\right)\right)\Leftrightarrow(c_i)_{i=1}^l\in\bigcap_{i=1}^lD_i^\complement.$$

The conjecture is motivated by the decomposition of finite-dimensional left-Hilbert modules into direct sums of $\mathbb{F}^m\otimes\mathbb{F}^n$ and of their underlying finite-dimensional $C^*$-algebras into direct sums of matrix algebras. The corresponding results are proved for finite-dimensional Hilbert spaces and finite-dimensional $C^*$-algebras, but remain conjectural for finite-dimensional left-Hilbert modules.