Compatibility of the vertical-collar filtration with the s-dot operad

Let $M$ be the ambient symplectic manifold and let $\operatorname{Lag}(M)$ denote its stable $\infty$-category of Lagrangian branes. The $s$-dot construction gives a colored planar $\infty$-operad, whose colors are equivalence classes of objects and whose operations are filtrations with prescribed associated gradeds. Vertically collared branes with $N=k$ and $N'=1$ likewise form a colored planar $\infty$-operad, with $X_1'$ at the root and $X_k,\ldots,X_1$ at the leaves.

Compatibility conjecture. The filtration of the vertical-collar theorem defines a map of colored planar $\infty$-operads to the colored planar $\infty$-operad obtained from the $s$-dot construction of $\operatorname{Lag}(M)$.

This conjecture asserts that the geometric filtration construction is compatible with the operadic structure of the $s$-dot construction. The source gives no resolution, so the claim remains open.