Algebraic overtwistedness conjecture for positive-braid rainbow closures

Let $\Lambda$ be a non-trivial Legendrian knot that is the Legendrian rainbow closure of a positive braid. For $k\in {\mathbb N}_+$, let $(S^{3}_{1/k}(\Lambda),\xi_{1/k}(\Lambda))$ denote the contact $1/k$-surgery on $\Lambda$. Positive-braid rainbow-closure conjecture. The contact manifold $(S^{3}_{1/k}(\Lambda),\xi_{1/k}(\Lambda))$ is algebraically overtwisted and tight for every $k\in {\mathbb N}_+$. The preceding theorems establish this conclusion for non-trivial Legendrian positive torus knots with maximal Thurston–Bennequin invariant and for the stated families of positive braids; the conjecture proposes it for all non-trivial Legendrian rainbow closures of positive braids.