Conjecture on exponent-one distal cell decompositions for locally modular geometric structures

Let $\mathcal{M}$ be a distal locally modular geometric structure, and let a distal cell decomposition refer to the cell-decomposition notion for definable families used in the paper. Its exponent is the exponent governing the growth of the number of cells as a function of the size of the parameter set.

Exponent-one locally modularity conjecture. Every distal locally modular geometric structure admits distal cell decompositions of exponent $1$.

The conjecture is supported by the cases established in the paper: the $o$-minimal case and the linear reduct of $\mathbb{Q}_p$. The general assertion remains open in the source.