Integration conjecture for generalized virtual-knot theories

Consider virtual-knot theories with skeleton equal to a circle (round), a line (long), or a line with all real crossings descending; with R2 and R3 treated in the standard, braid-like, or R2-only way; and with or without R1. In each theory, finite type invariants are defined by the virtual-knot difference relation, and a degree $n$ weight system is a weight system of degree $n$ for the corresponding arrow-diagram theory. Generalized virtual-knot integration conjecture. For each of these choices and every natural number $n$, every degree $n$ weight system comes from a type $n$ invariant. The assertion asks whether integration holds simultaneously for the 18 virtual-knot theories obtained from the listed choices. The surrounding text presents it as a broad question across these variants; the supplied material gives no proof or resolution.