Bödigheimer compactification conjecture for unoriented string diagrams

Let $\mathcal{SD}^{\mathrm{u}}$ be the space of unoriented string diagrams, let $\mathcal{SD}^{\mathrm{u}}/\sim$ be its quotient by the stated equivalence relation, and let $\mathcal{H}$ be Bödigheimer's harmonic compactification of the moduli space of Riemann surfaces with parameterized boundary. Bödigheimer compactification conjecture.

$$\mathcal{SD}^{\mathrm{u}}/\sim\simeq\mathcal{H}.$$

The conjecture proposes a compactified string-diagram model for the moduli space and connects the construction to Bödigheimer's harmonic compactification. The source does not report a resolution.