Paszkiewicz's conjecture on products of decreasing positive contractions

Let $H$ be a separable infinite-dimensional Hilbert space, and let $T_1\ge T_2\ge\dots$ be a sequence of positive linear contractions on $H$. Define

$$S_n:=T_nT_{n-1}\cdots T_1.$$

Paszkiewicz's conjecture. The sequence $(S_n)$ converges strongly.

This conjecture concerns strong convergence of products of a decreasing sequence of positive contractions. The paper proves the conjecture; the stronger equivalent formulation is that, if $T:=\lim_{n\to\infty}T_n$ in the strong operator topology and $P:=1_{\{1\}}(T)$, then $S_n$ converges to $P$ in the $*$-strong topology.