Is the local-density Ramsey exponent $c(p,q)$ strictly increasing in $q$? (Erdős #667)

Fix integers $p,q\geq 1$. Let $H(n;p,q)$ be the largest $m$ such that every graph on $n$ vertices in which every set of $p$ vertices spans at least $q$ edges must contain a complete graph $K_m$. Define $$c(p,q)=\liminf_{n\to\infty}\frac{\log H(n;p,q)}{\log n}.$$ Is $c(p,q)$ a strictly increasing function of $q$ on the range $1\leq q\leq \binom{p-1}{2}+1$? That is, does $c(p,q)<c(p,q+1)$ hold throughout this range for every fixed $p$?

Acceptance. FULLY RESOLVES: a complete proof that $c(p,q)$ is strictly increasing in $q$ on $1\leq q\leq \binom{p-1}{2}+1$ for every fixed $p$ (equivalently $c(p,q)<c(p,q+1)$ throughout), machine-checkable (Lean) preferred or a full written proof; OR a disproof exhibiting specific $p$ and $q$ in range with $c(p,q)=c(p,q+1)$, together with a proof of that equality (matching upper and lower exponent bounds). ADVANCES: prove strict monotonicity for a specific fixed $p$ (e.g. $p=4$ or $p=5$) across all admissible $q$, with proof; OR determine $c(p,q)$ exactly, or improve its best known upper or lower bound, for a specific pair $(p,q)$ beyond the values recorded here ($c(p,1)\in[\tfrac{1}{p-1},\tfrac{2}{p+1}]$, $c(p,\binom{p-1}{2})\leq 1/2$, $c(p,\binom{p-1}{2}+1)=1$), with proof; OR compute $H(n;p,q)$ exactly for a new range of small $n,p,q$ via exhaustive/SAT search, giving certified numerical bounds on $c(p,q)$ with reproducible code. Deliver a proof, a counterexample with proof, or improved bounds / certified small-case values.