Kapovich–Weidmann conjecture on higher stabilization for Nielsen equivalence

Let $k\ge m\ge 2$, and let

$$G=\langle a_1,\ldots,a_k,b_1,\ldots,b_m\mid a_i=u_i(\bar b),\ b_j=v_j(\bar a)\text{ for }1\le i\le k,\ 1\le j\le m\rangle.$$

Assume the tuples of defining words lie in the generic subsets specified in the construction and satisfy $|u_1|=|v_1|$.

Kapovich–Weidmann higher-stabilization conjecture. For every $1\le t<m$, the generating $(k+t)$-tuple $(a_1,\ldots,a_k,1,\ldots,1)$ is not equivalent to a $(k+t)$-tuple of type $(b_1,\ldots,b_{t+1},g_1,\ldots,g_{k-1})$.

This strengthens the preceding one-stabilization obstruction: fewer than $m$ stabilizations should not suffice to relate the generating tuples. The notation for the generic subsets and the precise equivalence relation is inherited from the surrounding construction.