Separated presentation adjunction for causal and triangular Frobenius presentations

Let $\mathbf{Caus}$ be the category of finite acyclic directed graphs equipped with a chosen topological order, with order-preserving graph homomorphisms or refinements as morphisms. Let $\mathbf{TangFrob}_{\mathrm{tri}}$ be the category of Frobenius Markov objects with tangent intervention fields $v_1,\dots,v_d$ and a filtration

$$0=\mathfrak g_0\subset \mathfrak g_1\subset\cdots\subset\mathfrak g_d, \qquad \mathfrak g_i=\operatorname{span}\{v_1,\dots,v_i\},$$

satisfying $[v_i,v_j]\in\mathfrak g_{>\max(i,j)}$, where $\mathfrak g_{>m}$ is the span of intervention directions strictly later than $m$. Let $F:\mathbf{Caus}\to\mathbf{TangFrob}_{\mathrm{tri}}$ be the geometric-realization functor, and let $\mathbf{TangFrob}_{\mathrm{sep}}\subseteq\mathbf{TangFrob}_{\mathrm{tri}}$ be the separated subcategory defined by the $\gamma$-separated visible-stratum hypothesis, strengthened so that the triangular filtration and nonzero residuals are invariant under Frobenius-compatible changes of presentation. Separated presentation adjunction. On $\mathbf{TangFrob}_{\mathrm{sep}}$, graphical extraction is functorial and the realization functor admits a right adjoint

$$F:\mathbf{Caus}\rightleftarrows\mathbf{TangFrob}_{\mathrm{sep}}:G.$$

Moreover, the unit $\eta:\mathrm{Id}_{\mathbf{Caus}}\Rightarrow GF$ is an isomorphism, while the counit $\epsilon:FG\Rightarrow\mathrm{Id}_{\mathbf{TangFrob}_{\mathrm{sep}}}$ is the triangularization map comparing each separated tangent Frobenius object with the object generated by its extracted acyclic presentation. The conjecture formalizes when tangent Frobenius data determine an ordered graphical presentation: separation should make extraction functorial, and realization followed by extraction should recover the original presentation. The source does not provide a resolution, so the status remains open; the precise strengthened separation condition and the behavior of the counit require verification.