Borisov-type criticality conjecture for positively curved surfaces

Let $\Sigma$ be a $2$-dimensional compact Riemannian manifold, possibly with boundary, with positive Gauss curvature. A $C^{1,\alpha}$ isometric embedding is an isometric embedding of class $C^{1,\alpha}$, and an image is locally convex if for every $p\in\Sigma$ there is a neighborhood $U$ such that $v(U)$ is convex. Borisov-type criticality conjecture.

(a) For any $\alpha>\frac{1}{2}$, the image of any $C^{1,\alpha}$ isometric embedding $v$ of $\Sigma$ in $\mathbb R^3$ is locally convex.

(b) For any $\alpha<\frac{1}{2}$, there is a $C^{1,\alpha}$ isometric embedding $v$ of $\Sigma$ in $\mathbb R^3$ which is not locally convex; in fact, any short embedding can be uniformly approximated by $C^{1,\alpha}$ isometric embeddings. This conjecture would give a stronger form of the criticality of the Hölder exponent $\frac{1}{2}$ for isometric embeddings of positively curved surfaces. The supplied text reports no resolution.