Off-diagonal semi-algebraic Ramsey bound

For integers $k\geq 3$, dimension $d$, complexity $t$, and $s$, let $R_k^{d,t}(s,n)$ be the least integer $N$ such that every sequence of $N$ points in $\mathbb{R}^d$ and every $k$-ary semi-algebraic relation of complexity at most $t$ contains either $s$ points all of whose induced $k$-tuples belong to the relation or $n$ points none of whose induced $k$-tuples belong to it. Off-diagonal semi-algebraic Ramsey conjecture. For fixed $k\geq 3,d,t$, and $s$, there is a constant $c=c(k,d,t,s)$ such that

$$R_k^{d,t}(s,n)\leq \operatorname{twr}_{k-2}(n^c).$$

This would improve the general tower-height upper bound by one level in the off-diagonal case. The paper presents it as a stronger expected bound, with the crucial $k=3$ case conjectured separately in polynomial form.