The answer is yes for all dimensions.

An old theorem by Fenchel states that for a compact set in $\mathbb{R}^n$ every point in the convex hull can be either written as a convex combination of at most $n$ points or be separated by a hyperplane. Since $M$ is path-connected, a separation is not possible.

But there are also more modern theorems of this kind available. A good starting point is recent work by Barany & Karasev. Their work also implies an affirmative answer to your question.

The answer is yes for all dimensions.

An old theorem by Fenchel states that for a compact set $K$ in $\mathbb{R}^n$ every point in the convex hull can be either written as a convex combination of at most $n$ points or $K$ can be separated by a hyperplane. Since $M$ is path-connected, such a separation is not possible.

But there are also more modern theorems of this kind available. A good starting point is recent work by Barany & Karasev. Their work also implies an affirmative answer to your question.