Day, Falgas–Ravry and Treglown's generalized Turán pattern conjecture

Let $a, r, s, d \in \mathbb{Z}_{\geq 0}$ with $a,r\geq 2$, $a>d$, and $s\geq (r-1)(d+1)+2$. Let $\mathbf{r}$ denote the $(d+1)$-tuple

$$\mathbf{r}=(r-1,0,0,\dotsc,0,1).$$

Day, Falgas–Ravry and Treglown's conjecture. For all $n$ sufficiently large,

$$\operatorname{ex}_\Pi\bigl(n,s,\Sigma_s(\mathrm{T}[\mathbf{r};a])\bigr)=\Pi_n(\mathrm{T}[\mathbf{r};a]).$$

This generalizes Mubayi and Terry's conjecture for $(s,q)=(4,a\binom{4}{2}+3)$. The claim is asymptotically true in the base case and, more generally, for sufficiently large $a$, but the full statement remains open.