Exact online size Ramsey number of a triangle versus a path

Let $C_3$ be the triangle and let $P_n$ be the path on $n$ vertices. Write $\tilde{r}(C_3,P_n)$ for the minimum number of rounds in the online size Ramsey game in which Builder forces either a red copy of $C_3$ or a blue copy of $P_n$.

Conjecture on the triangle–path online size Ramsey number.

$$\tilde{r}(C_3,P_n)=3n-4\text{ for every } n\ge 3.$$

The paper proves the upper bound $\tilde{r}(C_3,P_n)\le 3n-4$ and gives a lower bound with asymptotic coefficient $\varphi+1$, where $\varphi$ is the golden ratio. The authors state that they believe the upper bound is sharp for every $n\ge3$, so the equality remains open.