  Paszkiewicz's conjecture asserts that given a decreasing sequence $T_1\ge
T_2\ge \dots$ of positive contractions on a separable infinite-dimensional
Hilbert space $H$, the product $S_n=T_nT_{n-1}\cdots T_1$ converges in the
strong operator topology. In these notes, we give an equivalent, more precise
formulation of his conjecture. Moreover, we show that the conjecture is true
for the following two cases: (1) $1$ is not in the essential spectrum of $T_n$
for some $n\in \mathbb{N}$. (2) The von Neumann algebra generated by $\{T_n\mid
n\in \mathbb{N}\}$ admits a faithful normal tracial state. We also remark that
the analogous conjecture for the weak convergence is true.
