Criticality conjecture for isometric embeddings in arbitrary codimension

Let $\Sigma_a$ be the polar cap and let $\mathscr{I}^\alpha_1(\Sigma_a)$ denote the class of $C^{1,\alpha}$ isometric embeddings of $\Sigma_a$ into the relevant codimension-one Euclidean space, with distinguished coordinate functions $Y$ and $Z$. Criticality conjecture. For any $\alpha < \frac{1}{2}$ and any $0<a<1$ there is $v\in \mathscr{I}^\alpha_1(\Sigma_a)$ such that

$$\langle Y,Z\rangle>a.$$

This conjecture asserts that the codimension appearing in the low-regularity construction is not geometrically meaningful and that the same conclusion should hold in every codimension. It is motivated by the expected criticality of the Hölder exponent $\frac{1}{2}$; the supplied text gives no resolution.