  AGT conjecture reveals a connection between 4D $\mathcal{N}=2$ gauge theory
and 2D conformal field theory. Though some special instances have been proven,
others remain elusive and the attempts on its full proof never stop. When the
$\Omega$ background parameters satisfy $-\epsilon_1/\epsilon_2\equiv \beta =1$,
the story simplifies a bit. A proof of the correspondence in the case of
$A_{1}$ gauge group was given in 2010 by Mironov et al., while the $A_{n}$
extension is verified by Matsuo and Zhang in 2011, with an assumption on the
Selberg integral of $n+1$ Schur polynomials. Then in 2020, Albion et al.
obtained the rigorous result of this formula. In this paper, we show that their
result is equivalent to the conjecture on Selberg integral of Schur
polynomials, thus leading to the proof of the $A_{n}$ case at $\beta=1$. To
perform a double check, we also directly start from this formula, and manage to
show the identification between the two sides of AGT correspondence.
