Must large chromatic number with no K_t force two anticomplete c-chromatic subgraphs? (Erdős #1111)

For a finite graph $G$, let $\chi(G)$ be its chromatic number and $\omega(G)$ its clique number (largest number of pairwise adjacent vertices); for a vertex set $S$, $\chi(S)$ is the chromatic number of the induced subgraph $G[S]$. Two disjoint vertex sets $A,B$ are anticomplete if there is no edge between them. Conjecture (El Zahar–Erdős): for all integers $t,c\geq 1$ there exists $d\geq 1$ such that every graph $G$ with $\chi(G)\geq d$ and $\omega(G)<t$ contains anticomplete sets $A,B$ with $\chi(A)\geq\chi(B)\geq c$. Prove or disprove this, and estimate the least valid $d=d(t,c)$.

Acceptance. FULLY RESOLVES: prove the conjecture in full — for all $t,c$ the constant $d(t,c)$ exists (ideally with an explicit bound) — OR disprove it by exhibiting, for some $t,c$, graphs of arbitrarily large chromatic number with $\omega<t$ having no two anticomplete sets both of chromatic number $\geq c$. Complete proof (Lean/Coq preferred, otherwise a full written proof). ADVANCES, any of: (a) prove existence of $d(t,c)$ in the first open regime $c=4$ (any $t$), with proof; (b) determine a new exact value of $d(t,c)$ beyond the known $d(2,2)=2$, $d(3,2)=4$, $d(4,2)=5$, $d(3,3)\leq 8$, delivering an extremal $K_t$-free graph with $\chi=d(t,c)-1$ and no two anticomplete $c$-chromatic parts (machine-checkable) plus a certificate that the value is tight; (c) improve any upper bound on $d(t,c)$ stated in the background, strictly better than the best bound stated above, with proof; (d) strengthen the Nguyen–Scott–Seymour min-degree result toward the full $\chi(A)\geq c$ conclusion. Deliver the proof, or the extremal graph + verification/exhaustiveness certificate.