Irrationality conjecture for the roots associated with an infinite family of horospherical cones

For each integer $r\geq 5$, define

$$Q(t,r)=\int_{\frac{-4}{2+t}}^{\frac{2r-6}{1+t}}p(4+(2+t)p)^2(4r-8-tp)^2(2r-6-(1+t)p)^{2r-7}(2r-2+p)^{2r-7}\,dp.$$

The equation $Q(t,r)=0$ has a unique positive root. Irrationality conjecture. For all $r\geq 5$, this unique positive root is irrational. This is presented as an algebro-arithmetic reformulation of the regularity conjecture for the family $\operatorname{SL}_{r}/\operatorname{S}(\operatorname{GL}_2\times\operatorname{GL}_{r-2})$; the source attributes the polynomial characterization of the K-stability condition to previous work, while the irrationality remains conjectural.