Turán numbers of ordered tight hyperpaths

An ordered hypergraph is a hypergraph $G$ whose vertex set $V\left(G\right)$ is linearly ordered. We find the Turán numbers for the $r$-uniform $s$-vertex tight path ${\overrightarrow{P}}_s^{\left(r\right)}$ (with vertices in the natural order) exactly when $r\le s<2r$ and $n$ is even; our results imply $\overrightarrow{\mathrm{ex}}(n,{\overrightarrow{P}}_s^{\left(r\right)})=(1-\frac{1}{2^{s-r}}+o\left(1\right))\left(\frac{n}{r}\right)$ when $r\le s<2r$. When $s\ge 2r$, the asymptotics of $\overrightarrow{\mathrm{ex}}(n,{\overrightarrow{P}}_s^{\left(r\right)})$ remain open. For $r=3$, we give a construction of an $r$-uniform $n$-vertex hypergraph not containing ${\overrightarrow{P}}_s^{\left(r\right)}$ which we conjecture to be asymptotically extremal.