A remark on the Brown-Susskind conjecture

In the spirit of the Brown-Susskind conjecture, proven in \cite{haf} and \cite{li}, we study the growth of the dimension of the set of $n$-qubit unitaries which can be obtained as a product of a fixed number of $2$-qubit gates from a fixed set of pairs of qubits. We show that this dimension increases strictly when one more pair is added, for at least some choice of the $2$-qubit pairs at each step.