 Solution

Assume the spheroid is homogeneous, as implicit in the stated moments. Put $s=\sqrt{1-e^2}$ and map a uniform solid sphere by $(X_1,X_2,X_3)\mapsto (X_1,X_2,s{X}_3)$, preserving total mass by adjusting density. For the sphere, symmetry and its moment $2M{a}^2/5$ give $\int X_i^{2}dm=M{a}^2/5$ for each coordinate. The transformed squared-coordinate integrals are consequently $M{a}^2/5,M{a}^2/5,s^{2}M{a}^2/5$. Reflection symmetry makes the off-diagonal inertia entries vanish. ThusThe original PDF gives the correct solid-sphere factor $2/5$, rather than the converted TeX's $2/3$.