A tight chromatic bound in terms of triangle count and local triangle bound

Let $G$ be a graph with $n$ vertices, $t$ triangles, and local triangle bound $y$. Here, the local triangle bound $y$ is an upper bound on the number of triangles containing any given vertex. The conjectured tight bound. One should have

$$\chi(G) \lesssim \frac{t^{1/3}}{\log^{2/3}(t^2/y^3)}+\sqrt{\frac{n}{\log n}}.$$

This would strengthen the stated theorem, imply the proposition there as the special case $t\leq ny$, and match the given lower bound; its status is open.