Is the free abstract group residually of rank d > 2? Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements. 
Is there a finite nontrivial word $w = w(x_1, \dots, x_n)$ which is trivial on $\mathcal{F}_d$? 
That is, can it be that for any choice of $G \in \mathcal{F}_d$ and $y_1, \dots, y_n \in G$ we have $w(y_1, \dots, y_n) = 1$.
Alternatively: Let $F$ be a free group of rank $\aleph_0$. Is it possible that the intersection of all finite index subgroups $N \lhd F$ with $F/N \in \mathcal{F}_d$ is nontrivial? 
I am also interested in the analogous question for $p$-groups.
 A: No, there is not such a word, by the following two facts.


*

*The two elements
$$A=\left(\begin{array}{cc}1&2\\0&1\end{array}\right),\quad B=\left(\begin{array}{cc}1&0\\2&1\end{array}\right)$$ of $\text{SL}_2(\mathbf{F}_p)$ satisfy no nontrivial relation of length less than $c\log p$.

*Every subgroup of $\text{SL}_2(\mathbf{F}_p)$ is generated by at most 2 elements.


For 1, there is a well known ping-pong argument which proves that $A$ and $B$ generate a free subgroup of $\text{SL}_2(\mathbf{Z})$. The argument can be reproduced in $\text{SL}_2(\mathbf{F}_p)$ for words which are not too long.
For 2, the subgroups of $\text{SL}_2$ are well understood. Refer for instance to Theorem 6.17 in Suzuki's book:
Suzuki, Michio. Group theory. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 247. Springer-Verlag, Berlin-New York, 1982. xiv+434 pp. ISBN: 3-540-10915-3 MR0648772 
A: As I said in a comment on your other questions, there is no such word.  Indeed, for any $w\in F_n$, there is $u\in F_{n-1}$ so that $w$ survives under the retraction  $F_n\to F_{n-1}$ that sends the last generator to $u$.
