A dimension bound for circle actions with an odd number of fixed points

Let $S^1$ act on a compact oriented manifold $M$ with $k$ fixed points, where $k$ is odd.

Dimension-bound conjecture. The dimension of $M$ is bounded above by a function of $k$; more precisely,

$$\dim M \leq 8(k-1).$$

The conjecture is motivated by examples with three fixed points on $\mathbb{CP}^2$, $\mathbb{HP}^2$, and $\mathbb{OP}^2$, in dimensions $4$, $8$, and $16$, respectively, and by the nonexistence result in dimension $12$. It proposes that odd numbers of fixed points impose an upper bound on the dimension.