Maehara's strengthened common-point conjecture for unit spheres

Let $\mathfrak{F}$ be a family of at least $n+2$ distinct $(n-1)$-dimensional unit spheres in $\mathbb R^n$, where $n\geq 3$. Suppose that any $n+1$ spheres in $\mathfrak{F}$ have a point in common.

Maehara's conjecture. All the spheres in $\mathfrak{F}$ have a point in common.

This strengthens Maehara's theorem by lowering the required family size from $n+3$ to $n+2$. The conjecture is presented as a stronger version of that theorem, and no resolution is given here.