Asymmetric infinite sumsets in large sets of integers

We show that for any set $A\subset \mathbb{N}$ with positive upper density and any $\mathrm{\ell },m\in \mathbb{N}$, there exist an infinite set $B\subset \mathbb{N}$ and some $t\in \mathbb{N}$ so that $\{m{b}_1+\mathrm{\ell }{b}_2\text{\hspace{0.17em}⁣}:b_1,b_2\in B\text{ and }{b}_1<b_2\}+t\subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{m{b}_1+\mathrm{\ell }{b}_2\text{\hspace{0.17em}⁣}:b_1,b_2\in B\text{ and }{b}_1\le b_2\}$, for infinite $B\subset \mathbb{N}$ and prove that any set $A\subset \mathbb{N}$ with lower density $\underset{‾}{d}(A)>1\mathrm{/}2$ contains such configurations up to a shift. We show that the value $1\mathrm{/}2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.