Asymptotic extremal conjecture for ordered 3-uniform tight paths

Let $f(n,k)$ denote the maximum number of edges in an $n$-vertex ordered 3-uniform hypergraph avoiding the ordered tight-path configuration with parameter $k$, and let $P^{(3)}_s$ be the ordered 3-uniform tight path on $s$ vertices. Define

$$a=\left\lfloor\frac{(k+1)^2}{4}\right\rfloor,$$

and, equivalently,

$$b=\left\lfloor\frac{(s-1)^2}{4}\right\rfloor.$$

Asymptotic extremal conjecture for ordered 3-uniform tight paths. The constructions described in the source are asymptotically optimal:

$$f(n,k)=\left(1-\frac{1}{a}+o(1)\right)\binom{n}{3}.$$

Equivalently,

$$\mathrm{ex}_<(n,P^{(3)}_s)=\left(1-\frac{1}{b}+o(1)\right)\binom{n}{3}.$$

The preceding constructions give the corresponding lower bounds, so the conjecture concerns matching asymptotic upper bounds for all relevant parameters.