Here are the two versions of Arnold's conjecture on Hamiltonian orbits:

Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate Hamiltonian. Then the number of $1$-periodic orbits of the vector field $X_H$ defined by $\omega(X_H, \cdot) = dH$ is bounded below by

the sum of the rational Betti numbers of $M$ (

weak version)the minimum number of critical points of a Morse function on $M$ (

strong version).

The weak version has been settled using Floer homology. My question is:

Has anyone made any progress on the strong Arnold conjecture?

I have asked a couple of experts in person, but they didn't know of anything (see also Tim Perutz's answer to this question).

Also, I wonder if anyone can tell me the right intuition for the strong conjecture. The only idea I have for why the strong conjecture should be true is that when $H$ is time-independent, each critical point of $H$ is a $1$-periodic orbit of $X_H$.

**Edit about the version of the conjecture without assuming $H$ nondegenerate:** Thanks to Thomas's comment below I looked up Arnold's original statement of the conjecture and realized that it is different than the one I wrote above (which I have never seen in print). Call the conjecture above the nondegenerate Arnold conjecture; here is the original, "possibly degenerate" Arnold conjecture from *Mathematical methods of classical mechanics*:

Let $(M, \omega)$ be a closed symplectic manifold, and let $H: \mathbb{R}/\mathbb{Z} \times M \to \mathbb{R}$ be a Hamiltonian. Then the number of $1$-periodic orbits of $X_H$ is bounded below by the minimal number of critical points of a smooth function on $M$.

There is also a weak version (not from Arnold's book) where the bound is replaced by $1$ plus the cuplength of $M$.

Some progress has been made toward the strong version of this conjecture. In "On analytical applications of stable homotopy", Yuli Rudyak proves that if $(M,\omega)$ is a closed symplectic manifold such that $\omega|_{\pi_2(M)} = 0$ and $c_1|_{\pi_2(M)} = 0$ and the Lusternik-Schnirelmann category of $M$ is $\dim M$, then the strong version of the "possibly degenerate" Arnold conjecture holds. In "On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture", John Oprea and Rudyak eliminate the LS hypothesis by showing that $\text{LS}(M) = \dim M$ whenever $M$ is a closed symplectic manifold such that $\omega|_{\pi_2(M)} = 0$.