Satellites and invariants of links

An invariant $v$ of $m$-component links is called asymptotic if there exists a $k$ such that whenever a link $L'$ is obtained from a link $L=(K_1,\dots,K_m)$ by replacing each knot $K_i$ with its $(p_i,q_i)$-cable for some $p_i$ and $q_i$, we have $v(L')=(p_1\cdots p_n)^kv(L)$.
The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from V. I. Arnold's program of finding topological lower bounds for the energy of a magnetic field: Does there exist an asymptotic finite type invariant of links in $S^3$ which is not a function of the pairwise linking numbers? We solve it affirmatively. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function $\Omega_L(x_1,\dots,x_n)$ expanded as a formal power series in Conway's variables $z_i=x_i-x_i^{-1}$.
If time permits, we may also discuss type $n$ invariants which are "asymptotic up to an invariant of type $n-1$", some asymptotic invariants which are not of finite type (particularly a certain modification of Milnor's $\bar\mu$-invariants), and applications to links of solenoids.