Given $B(x_1,\dots,x_n,y_1,\dots,y_n)=x'Ay$ a bilinear form with $rank(A)=r$ with $A\in\{0,1\}^{n\times n}$.

Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.

Denote $J\in\{0,1\}^{n\times n}$ as matrix with $1$s.

Is it true $$\lim_{n\rightarrow\infty}\mathsf{P}\Bigg(\mathsf{1_n}'(J-A)\mathsf{1_n}>\frac{(r-1)r^{\log_2r}}{\log_2r}\Bigg)\rightarrow1?$$