A Selberg integral type formula for an $\mathfrak{sl}_2$ one-dimensional space of conformal blocks

For distinct complex numbers $z_1,\dots,z_{2N}$, we give a polynomial $P(y_1,\dots,y_{2N})$ in the variables $y_1,\dots,y_{2N}$ which is homogeneous of degree $N$, linear with respect to each variable, $\mathfrak{sl}_2$-invariant with respect to a natural $\mathfrak{sl}_2$-action, and is of order $N-1$ at $(y_1,\dots,y_{2N})=(z_1,\dots,z_{2N})$.
We give also a Selberg integral type formula for the associated one-dimensional space of conformal blocks.