For every large enough $T$ given four integers $a,b,c,d$ with absolute value less than $T^2$ are there integers $w_1,x_1,y_1,z_1,w_2,x_2,y_2,z_2$ with absolute value less than $T$ such that $$w_1x_1+y_1z_1=a$$ $$w_2x_1+y_2z_1=b$$ $$w_1x_2+y_1z_2=c$$ $$w_2x_2+y_2z_2=d$$ holds?

If not for what fraction of $a,b,c,d$ does it fail?