Asymptotic bound for the triangle-saturated minimum-degree constant

For fixed $t$, define $c(t,3)$ by

$$\operatorname{sat}_{t}(n,3)=tn-c(t,3)$$

for all sufficiently large $n$. The preceding construction gives a lower bound of order $2^{t}t^{3/2}$ for $c(t,3)$. The asymptotic bound conjecture.

$$c(t,3)=O(2^{t}t^{3/2}).$$

This conjecture predicts that the construction is asymptotically optimal up to a constant factor. The source gives no resolution.