The NLB conjecture for nontrivial words

For a positive integer $d$, let $\operatorname{F}(X_1,\ldots,X_d)$ be the free group on $X_1,\ldots,X_d$, and let $w\in\operatorname{F}(X_1,\ldots,X_d)$ be a nontrivial word. For a finite group $G$, write $\lambda(G)$ for its nonsolvable length. We say that $w$ is nonsolvable-length-bounding if there is a function $f_w:(0,1]\rightarrow[0,\infty]$ such that, for every $\rho\in(0,1]$ and every finite group $G$, whenever some fiber of the word map $w_G:G^d\rightarrow G$ has proportion at least $\rho$, one has $\lambda(G)\leq f_w(\rho)$. NLB conjecture. Every nontrivial word is nonsolvable-length-bounding. This conjecture is due to Michael Larsen and is described as very challenging; even the case $\rho=1$, when $w$ is an identity of $G$, is unknown.