Proved. The status-defining source is Theorem 1.2 of Kra, Moreira, Richter and Robertson [KMRR24] (Commun. Amer. Math. Soc. 4 (2024), 480--494, refereed), which proves the statement for every set of positive upper Banach density, of which positive upper density along the intervals is the special case; the paper calls it Erdős's $B+B+t$ conjecture. The claim page is Kra, Moreira, Richter and Robertson (accepted on the refereed publication and the site's credit).