Semiproportional irreducible characters of the groups $\mathrm{Sp}_4(q)$ and $\mathrm{PSp}_4(q)$ for odd $q$

All pairs of semiproportional irreducible characters (in other terms, small $D$-blocks) of symplectic groups $\mathrm{Sp}_4(q)$ and $\mathrm{PSp}_4(q)$ are obtained for odd $q$. In particular, two earlier-known conjectures on such pairs of characters in an arbitrary finite group are verified for these groups.