Distinct $t$-smooth components in a window of length $t$: is $f(n,t)\gg t$? (Erdős #461)

For a threshold $t$, let $s_t(m)$ be the $t$-smooth component of $m$: the product, with multiplicity, of all primes $p\mid m$ with $p<t$. For integers $n$ and $t$, let $f(n,t)$ be the number of distinct values taken by $s_t(m)$ as $m$ ranges over the window $[n+1,\,n+t]$. Is it true that $$f(n,t)\gg t$$ uniformly, i.e. there is an absolute constant $c>0$ (independent of $n$ and $t$) with $f(n,t)\ge c\,t$ for all $n$ and $t$?

Acceptance. FULLY RESOLVES: a complete proof that $f(n,t)\gg t$ holds uniformly (removing the $\log t$ factor from the best bound $f(n,t)\gg t/\log t$ stated in the background), or a proof that no such absolute constant exists (a construction of families $(n,t)$ with $f(n,t)=o(t)$). ADVANCES: prove a uniform lower bound strictly stronger than $f(n,t)\gg t/\log t$ — for instance $f(n,t)\gg t/\log\log t$ or $f(n,t)\gg t/(\log t)^{1/2}$ — with a rigorous proof; or, computationally, evaluate $f(n,t)$ over a large stated range of $n$ and $t$ and report the smallest observed ratio $f(n,t)/t$ together with the window that attains it, as a reproducible data point for or against the conjecture. Deliver the proof or the computation code plus the tabulated extremal ratios and witnessing windows.