Naive gluing conjecture for the comparison map

Let $(S,M,A)$ be a graded marked surface with a full arc system $A$. Say that it has enough marked intervals when

$$S\backslash A=\bigsqcup_{k=1}^N(D^2_k,M_k)$$

and, for each $k$, the inclusion $\iota_k:(D^2_k,M_k)\to(S,M)$ is injective on connected components, equivalently

$$(\iota_k)_*: \pi_0(M_k)\to\pi_0(M)$$

is injective. Let $\gamma_A:\mathbf n(S,A)\twoheadrightarrow\operatorname{Alg}(S,A)$ be the comparison map determined by $\gamma_A(a)=a$ for every $a\in A$. Naive gluing conjecture. If $(S,M,A)$ has enough marked intervals, then $\gamma_A$ is an isomorphism. This refines the stated surjectivity theorem: the conjecture asserts that the naive algebra has no further relations beyond the prescribed gluing relations, while the source gives no resolution status.