The naive-algebra presentation conjecture for Fukaya Hall composition subalgebras

Let $(S,M)$ be a graded marked surface with a full arc system $A$, and suppose that $(S,M)$ has enough marked intervals, meaning that for each disk $(D_k^2,M_k)$ in the decomposition induced by $A$, the inclusion $ota_k:(D_k^2,M_k)\to(S,M)$ is injective on connected components. Let $\mathbf{n}(S,A)$ be the naive algebra obtained by gluing the disk algebras, let $\operatorname{Alg}(S,A)$ be the composition subalgebra, and let

$$\gamma:\mathbf{n}(S,A)\twoheadrightarrow\operatorname{Alg}(S,A)$$

be the surjective homomorphism induced by sending each arc generator to the corresponding arc. Naive-algebra presentation conjecture. If $(S,M)$ has enough marked intervals, then $\gamma$ is an algebra isomorphism. The map is known to be an isomorphism when $S$ is a disk, while the assertion for general marked surfaces with enough marked intervals remains conjectural.