Independence number of $K_r$-free graphs: is the AEKS $\frac{\log t}{t}n$ bound true for all $r$? (Erdős #802)

Fix $r\geq 3$. Is it true that every $K_r$-free graph on $n$ vertices with average degree $t$ contains an independent set (a set of pairwise non-adjacent vertices) of size $$\gg_r \frac{\log t}{t}\,n,$$ where the implied constant may depend on $r$? For triangle-free graphs ($r=3$) this $\frac{\log t}{t}$ dependence is a classical theorem; the conjecture asserts that the same order persists for every fixed forbidden clique size $r$. Prove it for all $r$, or find an $r$ for which it fails.

Acceptance. FULLY RESOLVES: prove or refute the conjecture. A proof must establish that for every fixed $r\geq 3$ there is a constant $c_r>0$ such that every $K_r$-free $n$-vertex graph of average degree $t$ has an independent set of size $\geq c_r\frac{\log t}{t}n$ (complete written proof, or machine-checkable formalization). A refutation must exhibit, for some fixed $r$, an explicit family of $K_r$-free graphs whose independence number is provably $o\big(\frac{\log t}{t}n\big)$, with the clique-freeness and independence number certified. ADVANCES: prove the conjectured bound for a new value of $r\geq 4$ under the genuine $K_r$-free hypothesis (not the stronger neighbourhood-chromatic condition), or improve the general lower bound beyond Shearer's $\frac{\log t}{\log\log(t+1)\,t}n$ stated in the background, with a full proof. Deliver the proof, or the counterexample family plus verification code.