Two questions on "Table problem on $\Bbb S^2$"

Question

The following conjecture is known as "Table problem on $\Bbb S^2$"

Conjecture (Table problem on $\Bbb S^2$): Suppose $x_1, x_2,x_3,x_4 \in\Bbb S^2 \subseteq \Bbb R^3$ are the vertices of a square that is inscribed in the standard $2$-sphere, and let $h : \Bbb S^2\to \Bbb R$ be a smooth function. Then there exists a rotation $\rho\in \rm{SO}(3)$ such that $h(\rho(x_1)) =\cdots= h(\rho(x_4))$.

Question 1: What is the meaning of "standard $2$-sphere"? topological $2$-sphere or geometrical?

Question 2: Is this conjecture open still?

Answer 1

This is not an answer. but a list of references which wouldn't have fit in a comment (and wouldn't have been as easy to maintain). This is a digest of my colleague (A. Fruchard) answer to my query. Notice that the function $h$ may be assumed to be only continuous in the following first two results.

1. The conjecture is true when the square has same center as the sphere: F. J. Dyson, «Continuous Functions Defined on Spheres», Annals of Mathematics, Second Series, Vol. 54, #3, 1951, pp. 534-536

2. The conjecture holds inside a convex domain $D\subset \mathbb S^2$ if $h$ vanishes on $\partial D$. The square's vertices lie in $D$: R. Fenn, «The table theorem», Bull. London Math. Soc., #2, 1970, pp. 73-76

3. Related result. Let $S$ be a $\frac{1}{3}$-Lipschitz surface in $\mathbb R^3$ such that no square has exactly $3$ vertices in $S$. Then $S$ is contained in a plane or a sphere. N. Chevallier & A. Fruchard, «On the square property for Lipschitz surfaces», Rev. Roumaine Math. Pures Appl. 50, #5-6, 2005, pp. 515–525.