Hamiltonicity conjecture for the face lattice of every polytope

Let $P$ be a polytope of dimension $d\geq 1$. Its face lattice $L(P)$ consists of all faces of $P$, ordered by inclusion, and $G(L(P))$ is the cover graph of this lattice: its vertices are the faces of $P$, with two faces adjacent when one is a subface of the other and their dimensions differ by $1$.

Hamiltonicity conjecture. The graph $G(L(P))$ has a Hamiltonian cycle.

Such a cycle would give a Gray code listing of all faces of $P$, in which consecutive faces differ by a single cover relation in the face lattice. The source presents this as a brave conjecture and notes that the question appears to be novel; the available context gives no resolution.