Weighted-average tail conjecture for independent random variables

Let $X$ and $Y$ be independent random variables on $\mathbb{R}_+$ that are not compactly supported, and let $\lambda\in(0,1)$. Weighted-average tail conjecture.

$$\limsup_{m\to\infty}\frac{\mathbb{P}(\lambda X+(1-\lambda)Y>m)}{\mathbb{P}(X>m)\mathbb{P}(Y>m)}=\infty.$$

This conjecture is the main proposed extension from identically distributed variables to merely independent variables, and would imply the stated high-dimensional weighted-average result in the non-identically distributed case. The source gives no resolution, so the conjecture remains open.