Let $f(n)$ denote the proposition "There exists some $k>1$ such that $$ \sum_{m=k}^{k+n-1}\tau(m) < \sum_{m=1}^n\tau(m) $$ where $\tau(m)$ is the sum of the divisors of $m$." (This is like the so-called second Hardy-Littlewood conjecture on $\pi(x)+\pi(y)$ vs. $\pi(x+y)$ in testing the initial interval, which has guaranteed good behavior, vs. arbitrary intervals, which can perhaps be chosen cleverly.)

$f$ seems like a natural object, so I imagine this is not a novel question. What is known about $f(n)$ in terms of truth, falsity, or conjecture? It seems natural to expect $f(n)$ to be true for at least some $n$, maybe all large $n$, just like in 2HL, but finding such an $n$ does not seem easy.

I've convinced myself that $f(n)$ is false for $n<1000,$ but I don't have a proof. It's easy for small $n$ but it gets increasingly inconvenient for larger $n$.