I was reading the nice survey on rigidity, focusing on tensegrities by Connelly, and I'd like to know the status and feedback about a question he asks:

Theorem 3.1 in the above, says that for cable/strut/bar finite frameworks, local flexibility (i.e. the existence of non-trivial continuous deformations which respect the tensegrity constraints) is equivalent to infinitesimal flexibility (namely, the conditions one obtains at the first jet level from the above: explicitly, there exist $p_i'$ "tangent vectors" at configuration points $p_i$, satisfying natural constraints such as $(p_i-p_j)\cdot(p_i'-p_j')\le 0$ if $\{p_i,p_j\}$ is a cable, and analogues for bars and struts).

The proof of one implication is trivial, the other implication follows by algebraic regularization.

Connelly asks for a continuous analogue of this result.

One can give an interpretation of this question by asking: is an isometric immersion of a Riemannian manifold in Euclidean space locally flexible (in the sense that there is a path of isometric immersions that passes through it, not coming from a path of isometries of ambient space) if and only if it is infinitesimally flexible (=condition obtained by taking the 1st jet, or tangent vectorfield, version of the above)?

I am curious if the above interpretations seems legit, and if some answers are known, especially in the setting of immersions of $C^{1,\alpha}$-regularity for $\alpha$ small (or zero). In that case, one is probably supposed to take weak derivatives in the definition of infinitesimal rigidity. Or alternatively, give another interpretation of the question.

The emphasis on regularity is motivated by the famous rigidity results by Cohn-Vossen / Pogorelov / Nirenberg / Borisov, which work at higher regularity. If all isometric immersions are rigid, then the question becomes less interesting. Flexibility has been proved at lower $\alpha$ by De Lellis - Inauen - Szekelihidi (amongst others) following a path started by Nash, and this makes the question perhaps more interesting in that setting. For a better discussion on this see this survey.