Assume that we have an $n$-tuple $S^0=(S^0_1,\dots,S^0_n)$ of commuting operators in a Hilbert space $H$ and another such an $n$-tuple $S^1$. Is it possible to connect these two $n$-tuples by a piecewise smooth homotopy $S(t)$ of commuting $n$-tuples having the length comparable with the distance from $S^0$ to $S^1$?
More precisely, is it possible to find piecewise smooth mapping $t\mapsto S(t)$, $0\le t\le 1$ such that
1) $S(0)=S^0$ and $S(1)=S^1$;
2) For any $t$ we have that $S(t)=(S_1(t),\dots,S_n(t))$ is an $n$-tuple of commuting operators;
3) The length of the curve $S(t)$ (in a reasonable metric) is majorized by $M\|S^0-S^1\|$, where $M$ is some constant not depending on $S^0$ and $S^1$?
A similar question for tuples of self-adjoint operators was asked 5 years ago (and it is still unanswered): Moving one family of commuting self-adjoint operators to another without losing commutativity on the way. My question is a version of that without the requirement of self-adjointness.
