Minimal-dimension conjecture for mapping class group actions on spheres

Let $S_g$ be a closed orientable surface of genus $g\geq 3$, let $\Mod(S_g)$ denote its mapping class group, and let $\Homeo^+(\mathbb{S}^n)$ denote the group of orientation-preserving homeomorphisms of the $n$-sphere. Minimal-dimension conjecture. For $g\geq 3$:

1. The minimal $n$ such that there exists a nontrivial homomorphism

$$\Mod(S_g)\to \Homeo^+(\mathbb{S}^n)$$

is $n=2g-1$.

2. The minimal $n$ such that there exists an injective homomorphism

$$\Mod(S_g)\to \Homeo^+(\mathbb{S}^n)$$

\nis $n=6g-7$.

These assertions propose that the symplectic representation gives the smallest-dimensional nontrivial sphere action, while the action on the space of projective measured foliations gives the smallest-dimensional faithful sphere action. The minimality of these geometric examples is not known in general.