Analyzing a Seneta's conjecture by using the Williamson transform

Considering slowly varying functions (SVF), Seneta in 2019 conjectured the following implication, for $\alpha\geq1$, $$\int_0^x y^{\alpha-1}(1-F(y))dy\textrm{ is SVF}\ \Longrightarrow\ \int_{[0,x]}y^{\alpha}dF(y)\textrm{ is SVF, as } x\to\infty,$$ where $F(x)$ is a cumulative distribution function on $[0,\infty)$. Complementary results related to this transform and particular cases of this extended conjecture are discussed.