Skew blow-up conjecture for triangle densities

Let $B=K_{a,b,c}$ be a blowup of the triangle, where $a,b,c$ are positive integers not all equal. Let $f_B(\gamma)$ denote the minimum possible $B$-density in a graph of triangle density $\gamma$.

Skew blow-up conjecture. There is a constant $c>0$ such that

$$f_B(\gamma)\leq \gamma^{\frac13(ab+bc+ac)+c}.$$

This predicts that the random graph does not minimize the density of a skewed blowup. The construction in the paper verifies related cases, including bounds for $K_{1,1,t}$ up to an $o(1)$ term, but the general assertion remains open in the supplied text.