Falgas–Ravry's flat-interval conjecture

Let $\mathrm{T}[(r);a]$ be the generalized Turán pattern with parameter $r$ and ambient edge multiplicity $a$, and let $\Sigma_s(\mathrm{T}[(r);a])$ denote its associated threshold. Falgas–Ravry's flat-interval conjecture. For every $r,a\in\mathbb{Z}_{\geq 1}$ and every $s\geq 2r+1$,

$$\operatorname{ex}_\Pi\bigl(s,\Sigma_s(\mathrm{T}[(r);a])\bigr)=\cdots=\operatorname{ex}_\Pi\bigl(s,\Sigma_s(\mathrm{T}[(r);a])+\lfloor (s-1)/r\rfloor-1\bigr).$$

In particular, for all $0\leq t\leq\lfloor (s-1)/r\rfloor-1$,

$$\operatorname{ex}_\Pi\bigl(s,\Sigma_s(\mathrm{T}[(r);a])+t\bigr)=a\left(\frac{a+1}{a}\right)^{(r-1)/r}.$$

The conjecture predicts flat intervals of the multiplicative extremal function. The cited source presents it as an open conjecture; the surrounding paper gives supporting intuition but does not resolve it.