Complete the missing configuration contributions for the 4-orthoplex first volumetric moment

Determine closed-form expressions for the configuration contributions v_4^{(1)}(O_4)_{III} and v_4^{(1)}(O_4)_{IV} in the decomposition of the mean 4-volume of a random 4-simplex selected uniformly from the 4-orthoplex O_4, thereby completing v_4^{(1)}(O_4).

Background

For the 4-orthoplex O_4, the authors decompose the first volumetric moment into configuration-wise contributions determined by the symmetry genealogy. They obtained explicit expressions for some configurations, but not for configurations III and IV.

They note that they reduced these two cases to explicit double integrals but were unable to evaluate them with available computational resources, leaving the completion of v_4^{(1)}(O_4) open.