Number of solutions of overdetermined quadratic polynomial equations

Question

Given $m$ linearly independent quadratic polynomials over the complex field in $n$ variables with $m>n$ and such that the number of zeros, say $N$, is finite, is there a known or conjectured strict upper bound on $N$? For almost complete intersections, it is conjectured that the strict upper bound is $2^{n-1}+2^{n-2}$.