Say I have a vector space $V$ over $\mathbb{C}$, generated by $2n$-letters, $(x_1,...,x_n,y_1,...,y_n)$. Let $C_d$ denote the commutators on $V$ of length $d$, and let $(B_1,...,B_r)$ denote a basis for $C_d$, consisting of basic commutators of length $d$, given in the letters $x_i$ and $y_j$ only. Let $A$ and $B$ be disjoint subsets of $\{1,...,r\}$, where $A$ contains all those indices which contain an odd number of $y_j$'s, and let $B$ denote its complement.
My question is, can we bound $\lim_{d\rightarrow \infty} |A|/r$ away from $1$?
Edit: I conjecture the limit should be $1/2$, and the statement is trivial for odd $d$, since we can define the involution $x_i\leftrightarrow y_i$, which replaces our basis, swaps $A$ and $B$, however the cardinalities of $|A|$ and $|B|$ are basis independent.
