The axiomatization conjecture for extended directories

Let $K_0$ be the coefficient field, and let $D_\bullet=(D_0,D_1,D_2,\ldots)$ be a structure with a poset structure on each $D_i$. For every $m\times n$ matrix $\mu$ over $K_0$, suppose there are connecting maps

$$\mu^*:D_m\to D_n,\qquad \mu_*:D_n\to D_m.$$

Assume that each $D_n$ is a bounded modular lattice, $D_0$ is the trivial one-element modular lattice, the maps are order-preserving and functorial in $\mu$, with

$$(\mu\cdot\nu)^*=\mu^*\circ\nu^*,\qquad (\mu\cdot\nu)_*=\nu_*\circ\mu_*,\qquad (I_n)^*=\operatorname{id}_{D_n},\qquad (I_n)_*=\operatorname{id}_{D_n},$$

and that they satisfy the Galois connection

$$\mu_*(x)\le y\iff x\le\mu^*(y).$$

Axiomatization conjecture for extended directories. Then $D$ is an extended directory: there is an object $A$ in a $K_0$-linear abelian category such that

$$D\cong\operatorname{Dir}^+(A).$$

This proposed axiomatization would characterize extended directories by lattice-theoretic, functorial, and Galois-connection axioms. The supplied text presents it as a candidate axiomatization and gives no resolution.