The conjunction complexity asymptotic of self-correcting circuits for monotone symmetric functions with threshold $2$

It is stated that the conjunction complexity $L_k^{\&}(f^n_2)$ of monotone symmetric Boolean functions $f_2^n(x_1,\ldots,x_n)=\bigvee \limits_{1\leq i<j\leq n}x_i x_j$ realized by $k$-self-correcting circuits in the basis $B=\{\&,-\}$ asymptotically equals $(k+2)n$ for growing $n$ when the price of a reliable conjunctor is $\geq k+2$.