Paracyclic compatibility of the vertical-collar filtration

Let $M$ be the ambient symplectic manifold, and consider the map from vertically collared branes to the colored planar $\infty$-operad obtained from the $s$-dot construction of $\operatorname{Lag}(M)$. Rotating collared ends below and above the zero section $\mathbb R\subset T^*\mathbb R$ is periodic up to a grading shift: repeating the operation $N+1$ times returns the same collared brane with shifted grading.

Paracyclic compatibility conjecture. The map is compatible with the paracyclic action on the $s$-dot construction.

The claim concerns compatibility between the grading-shift periodicity of the geometric construction and the paracyclic action on the categorical $s$-dot construction. The source gives no resolution, so it remains open.