Conjecture on the number of involution isomorphy classes in symplectic groups

Let $k$ be a field, let $n>2$, and let $C_i(2n,k)$ denote the number of isomorphy classes of involutions of Type $i$ in $\operatorname{SP}(2n,k)$, for $1\leqslant i\leqslant 4$. Let $k^*$ be the multiplicative group of $k$ and $(k^*)^2$ its subgroup of squares. Conjecture on the number of involution isomorphy classes.

$$\begin{aligned} &\text{If } n \text{ is odd, then } C_1(2n,k)=\frac{n-1}{2};\quad \text{if } n \text{ is even, then } C_1(2n,k)=\frac{n}{2}.\\ &\text{If } n \text{ is odd, then } C_2(2n,k)=0;\quad \text{if } n \text{ is even, then } C_2(2n,k)=\left|k^*/(k^*)^2\right|-1.\\ &C_3(2n,k)=1.\\ &C_4(2n,k)=\left|k^*/(k^*)^2\right|-1. \end{aligned}$$

The conjecture proposes the maximal numbers of isomorphy classes for all four involution types over an arbitrary field. The paper establishes these values for finite and real fields, while the general case is stated as an open direction; the supplied status evidence indicates that the conjecture is disproved, although no counterexample is specified here.