Algebraic torsion conjecture for contact manifolds containing glued Giroux domains

Let $(V,\xi)$ be a closed contact manifold containing a subdomain $N$ obtained by gluing and blowing down Giroux domains, as in Theorem~.

Choose any $c \in {\mathcal{O}}(N)$ and consider symplectic field theory with coefficients in

$${\mathbb{R}}[H_2(V;{\mathbb{R}}) / \ker c].$$

Algebraic torsion conjecture. Then $(V,\xi)$ has algebraic $1$-torsion, and it is also algebraically overtwisted if $N$ contains any blown down boundary components.

This proposes an algebraic symplectic-field-theoretic detection of the filling obstruction established for such Giroux-domain subdomains. The claim is presented as an expected result, and no proof or resolution is supplied here.