Asymptotic extremal conjecture for path blowups

Let $a+b=r$, with $a,b\geq 1$, and let $P_\ell(a,b)$ be the $(a,b)$-blowup of a path of length $\ell$, where $\ell\in\{2k-1,2k\}$ and $\ell\geq 4$. For fixed $a,b,r,k$, define

$$\Psi_{k-1}(n,r):=\{E\subset [n]: |E|=r,\ E\cap [k-1]\ne\emptyset\}.$$

Path-blowup conjecture. The family $\Psi_{k-1}(n,r)$ gives the correct asymptotic of the Turán number in all the above cases.

The theorem preceding this conjecture proves the assertion for odd $r$ when $\ell$ is odd and for $a>b$ when $\ell$ is even; the remaining cases, namely even $\ell$ with $a\leq b$, are stated to be open.