Define higher arcsines as follows:
$$\arcsin(a) = \int_0^a \frac 1 { \sqrt{1-\alpha^2} } d\alpha$$
$$\arcsin(a,b,c) = \int_0^b \frac { \arcsin\left( a \beta c / v / w \right) } { \sqrt{1-\beta^2} } d\beta$$
$$\text{where}\; v = \sqrt{ 1-a^2-\beta^2 },\; w = \sqrt{ 1-\beta^2-c^2 }.$$
$$\arcsin(a,b,c,d,e) = \int_0^c \frac { \arcsin( x )\arcsin( y )+\arcsin( x, b \gamma d / v / w, y ) } { \sqrt{1-\gamma^2} } d\gamma$$
$$\text{where}\; v = \sqrt{1-b^2-\gamma^2},\; w = \sqrt{1-\gamma^2-d^2},\; x = \frac{a}{v} \sqrt{1-\gamma^2},\; y = \frac{e}{w} \sqrt{1-\gamma^2}.$$
Now define the orthoscheme volume:
$$V( a, b, c, d, e ) = \frac {\pi^3} {384} + \frac {\pi^2} {192} \left( \arcsin( a ) + \arcsin( b ) + \arcsin( c ) + \arcsin( d ) + \arcsin( e ) \right) + \frac \pi { 96 } \left( \arcsin( a ) \arcsin( c ) + \arcsin( a ) \arcsin( d ) + \arcsin( a ) \arcsin( e ) + \arcsin( b ) \arcsin( d ) + \arcsin( b ) \arcsin( e ) + \arcsin( c ) \arcsin( e ) + \arcsin( a, b, c ) + \arcsin( b, c, d ) + \arcsin( c, d, e ) \right) + \frac 1 {48} \arcsin( a, b, c, d, e ) .$$
Set the following in terms of parameter $u$:
$$\begin{array}{lll} a = 1-u^2 & b = 1+u^2 & d = 1+2 u^2 \\ e = 3-2 u^2 & f = 3+5 u^2 & g = 1+7 u^2 \\ h = 7-u^2 & i = 3+18 u^2-5 u^4 & j = 2-u^2 \\ k = 11-3 u^2 & l = 3+u^2 & m = 5-u^2 \\ n = 4-3 u^2 & p = 3-u^2 & q = 1+5 u^2 \end{array}.$$
Then for $0 \leq u < 1$, the orthant probability for $\Sigma$ is $\frac{6S}{\pi^3}$ where $S=$
$$\begin{array}{ll} - \: 4 \: V( u, \frac{-a}{\sqrt{d}}, \frac{u}{\sqrt{d}}, \frac{-2 u}{\sqrt{3 b}}, -\sqrt{\frac{a}{3b}} ) & - \: 2 \: V( u, \frac{-a}{\sqrt{d}}, \frac{u^2}{\sqrt{3 d}}, \frac{2\sqrt{2}}{3}, -\sqrt{\frac{a}{3f}} ) \\ - \: 2 \: V( \frac{1}{2}, \frac{-u}{2}, \sqrt{a}, \frac{4 u^2}{\sqrt{i}}, \frac{a}{\sqrt{3 i}}) & - \: 2 \: V( \frac{1}{2}, \frac{-u}{2}, \frac{m}{2 \sqrt{h}}, \frac{l}{2}\sqrt{\frac{a}{h i}}, u^2 m \sqrt{\frac{2}{f i}} ) \\ - \: 4 \: V( \frac{1}{2}, \frac{-3u}{2 \sqrt{d}}, \frac{a}{\sqrt{d q}}, -2 u^2 \sqrt{\frac{3}{b q}}, -\sqrt{\frac{a}{3 b}} ) & - \: 2 \: V( \frac{1}{2}, \frac{-3u}{2 \sqrt{d}}, \frac{a}{2 \sqrt{d h}}, \sqrt{\frac{6}{h}}, -\sqrt{\frac{a}{3 f}} ) \\ - \: 2 \: V( \frac{u}{2}, \frac{-\sqrt{3}}{2}, \frac{-a}{2 \sqrt{b}}, \frac{l u}{2 \sqrt{b p}}, -u \sqrt{\frac{6 a}{f p}} ) & - \: 2 \: V( \frac{u}{2}, \frac{-j}2, u \sqrt{\frac{a}{n}}, \frac{4 j u}{\sqrt{i n}}, \frac{a}{\sqrt{3 i} }) \\ - \: 2 \: V( \frac{u}{2}, \frac{-j}2,\frac{u^2 m}{2 \sqrt{3 b}}, \frac{j l}{2}\sqrt{\frac{a}{3 b i}}, u^2 m \sqrt{\frac{2}{f i}} ) & + \: 2 \: V( u, \frac{-a}{\sqrt{d}}, \frac{u^2}{\sqrt{3 d}}, \frac{l}{3 \sqrt{b}}, u^2 \sqrt{\frac{2 a}{3 b f}} ) \\ + \: 2 \: V( u, \frac{-a}{\sqrt{d}}, u^2 \sqrt{\frac{3}{d}}, \frac{a}{\sqrt{b}}, u^2 \sqrt{\frac{2}{b}} ) & + \: 2 \: V( \frac{1}{2}, \frac{-u}{2}, \frac{\sqrt{a}}{2}, \frac{\sqrt{3}}{2}, -u^2 \sqrt{\frac{2}{g}} ) \\ + \: 2 \: V( \frac{1}{2}, \frac{-u}{2}, \frac{\sqrt{a}}{2}, 2 \sqrt{\frac{2}{k}}, \frac{-a}{\sqrt{3 g k}}) & + \: 2 \: V( \frac{1}{2}, \frac{-u}{2}, \frac{m}{2 \sqrt{h}}, \sqrt{\frac{2 a}{h k}}, \frac{-m}{\sqrt{k f} }) \\ + \: 2 \: V( \frac{1}{2}, \frac{-3u}{2 \sqrt{d}}, \frac{1}{2}\sqrt{\frac{a}{d}}, \frac{1}{2}\sqrt{\frac{3 a}{b}}, u^2 \sqrt{\frac{2}{b}} ) & + \: 2 \: V( \frac{1}{2}, \frac{-3u}{2 \sqrt{d}}, \frac{a}{2 \sqrt{d h}}, \frac{l}{2}\sqrt{\frac{3}{b h}}, u^2 \sqrt{\frac{2 a}{3 b f}} ) \\ + \: 4 \: V( \frac{1}{2}, \frac{-1}{2}, \frac{-2 u}{\sqrt{q}}, \frac{1}{2}\sqrt{\frac{a}{q}}, \frac{-1}{2} ) & + \: 4 \: V( \frac{u}{2}, \frac{-1}{2}, -\sqrt{\frac{2}{3}}, -\sqrt{\frac{a}{3 f}}, -u \sqrt{\frac{2}{f}} ) \\ + \: 4 \: V( \frac{u}{2}, \frac{-1}{2}, \frac{-3}{2}\sqrt{\frac{a}{e}}, -2 u^2 \sqrt{\frac{2}{e f}}, -a \sqrt{\frac{3}{f g}} ) & + \: 4 \: V( \frac{u}{2}, \frac{-1}{2}, \frac{-3}{2}\sqrt{\frac{a}{e}}, \frac{-u}{2 \sqrt{e}}, -u \sqrt{\frac{6}{g}} ) \\ + \: 2 \: V( \frac{u}{2}, \frac{-\sqrt{3}}{2}, \frac{-1}{2}\sqrt{\frac{a}{d}}, \frac{-3u}{2}\sqrt{\frac{a}{d p}}, u \sqrt{\frac{2}{p}} ) & + \: 2 \: V( \frac{u}{2}, \frac{-\sqrt{3}}{2}, \frac{-a}{2 \sqrt{b}}, u^2 \sqrt{\frac{2}{b}}, \sqrt{\frac{3 a}{f}} ) \\ + \: 2 \: V( \frac{u}{2}, -\sqrt{a}, \frac{-u^2}{\sqrt{e}}, -2 \sqrt{\frac{2 a}{3 e}}, \frac{-1}{3}\sqrt{\frac{a}{g}} ) & + \: 2 \: V( \frac{u}{2}, -\sqrt{a}, \frac{-u^2}{\sqrt{e}}, -a \sqrt{\frac{3}{e}}, -2 u^2 \sqrt{{2}{g}} ) \\ + \: 2 \: V( \frac{u}{2}, -\sqrt{a}, \frac{-u}{\sqrt{n}}, -8 u \sqrt{\frac{a}{3 g n}}, \frac{1}{3}\sqrt{\frac{a}{g}} ) & + \: 2 \: V( \frac{u}{2}, -\sqrt{a}, -u^2 \sqrt{\frac{3}{d}}, -a \sqrt{\frac{a}{d g}}, 2 u^2 \sqrt{\frac{2}{g}} ) \\ + \: 2 \: V( \frac{u}{2}, \frac{-j}{2}, \frac{u^2}{2}\sqrt{\frac{a}{e}}, \frac{j}{2}\sqrt{\frac{3}{e}}, -u^2 \sqrt{\frac{2}{g}} ) & + \: 2 \: V( \frac{u}{2}, \frac{-j}{2}, \frac{u^2}{2}\sqrt{\frac{a}{e}}, 2 j \sqrt{\frac{2}{e k}}, \frac{-a}{\sqrt{3 g k} }) \\ + \: 2 \: V( \frac{u}{2}, \frac{-j}{2}, \frac{u^2 m}{2 \sqrt{3 b}}, j \sqrt{\frac{2 a}{3 b k}}, \frac{-m}{\sqrt{k f} }) & . \end{array}$$
For instance, for $u=\frac{1}{2}$, we have $\frac{6S}{\pi^3} \approx 0.11502383599812541648615657162020006115$.