Given $N\in\Bbb N$ such that $\prod_{i=1}^mp_i=N$ with $p_i$ being similar sized primes such that $p_i\neq p_j$ if $i\neq j$ where $m\in[1,\log\log N]$, consider $$r_4(N,[a,b])=|\{\alpha^2+\beta^2+\gamma^2+\delta^2=N:\alpha,\beta,\gamma,\delta\in[a,b]\cap\Bbb Z\}|.$$

What is $r_4(N,[2^{\frac{\log N-c}2},2^{\frac{\log N}2}])$ where fixed $c$ satisfies $0<c<\log\log N$?

Is this asymptotically $2^{\frac{\log N-c}{2m}}$?