Paszkiewicz conjecture on strong convergence of products of positive contractions

Determine whether, for every decreasing sequence T1 ≥ T2 ≥ ⋯ of positive linear contractions on a separable infinite-dimensional Hilbert space H, the product S_n = T_n T_{n-1} ⋯ T_1 converges in the strong operator topology.

Background

The conjecture concerns operator-theoretic behavior of products of positive contractions on a separable infinite-dimensional Hilbert space. Given a decreasing sequence T1 ≥ T2 ≥ ⋯, the limit T := lim_{n→∞} T_n exists in the strong operator topology, and the conjecture asks if the product S_n := T_n ⋯ T_1 always converges strongly.

Prior work established the conjecture in specific settings, such as when the generated von Neumann algebra is finite or when the operators have a uniform spectral gap at 1, and showed related facts such as lim_{n→∞} S_n^* = P in SOT. This paper claims to prove the conjecture in full generality and explores a generalized version under reordered products.