Conjecture on separating asymptotic weak Turán-goodness by blow-up containment

Let $F$ and $F'$ be graphs with chromatic number $r+1$. A graph $H$ is asymptotically weakly $F$-Turán-good if ${\mathrm{ex}}(n,H,F)=(1+o(1))\mathcal N(H,T)$ for some complete $r$-partite graph $T$. A graph $F'$ is contained in a blow-up of $F$ if it is a subgraph of a graph obtained by replacing each vertex of $F$ with an independent set and each edge with all edges between the corresponding sets. Blow-up separation conjecture. There exists a graph $H$ that is asymptotically weakly $F$-Turán-good but not asymptotically weakly $F'$-Turán-good if and only if $F'$ is not a subgraph of any blow-up of $F$. Furthermore, if $F$ has a color-critical vertex, then one can choose $H$ to be $F$-Turán-good. This conjecture extends the paper's separation results for non-bipartite forbidden graphs; the general assertion remains open.