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This is only a partial answer. The Beukers-Kolk-Calabi change of variables $$x_1=\frac{\sin{v_1}}{\cos{v_2}},\;\;x_2=\frac{\sin{v_2}}{\cos{v_3}},\ldots, \;x_{n-1}=\frac{\sin{v_{n-1}}}{\cos{v_n}},\;\;x_n=\frac{\sin{v_n}}{\cos{v_1}}$$ has the Jacobian $$\frac{\partial(x_1,\ldots,x_n)}{\partial(v_1,\ldots,u_n)}= 1-(-1)^n\,x^2_1x^2_2\cdots x^2_n.$$ Therefore you integral is a volume of the polytope $\delta_{2m+1}=\left \{(u_1,\ldots,u_{2m+1}): u_i\ge 0,\; u_i+u_{i+1}\le \pi/2 \right \}$. It remains to relate the volumes of the polytope $\delta_{2m+1}$ and $(2m+1)$-simplex $\Delta_{2m+1}$. See https://arxiv.org/abs/math/0101168