Existence of a second-order rigid but locally and continuously flexible framework

Let $(X,\|\cdot\|)$ be a normed space and let $(G,p)$ be a framework in $X$. The framework is second-order rigid if it admits no nontrivial second-order flex, and locally flexible or continuously flexible if it admits, respectively, a nontrivial local or continuous flex.

Existence conjecture. There exists a normed space with a second-order rigid framework that is locally and continuously flexible.

The conjecture concerns the remaining unknown implication between second-order rigidity and local or continuous rigidity in general normed spaces. The methods proving that implication in Euclidean normed spaces rely substantially on smoothness of the norm and on analytic flexes, so the question remains open for arbitrary normed spaces.