Cyman–Dzido–Lapinskas–Lo exact conjecture for the online Ramsey number of $P_4$

Let $P_t$ denote the path on $t$ vertices, and let $\tilde{r}(P_4,P_n)$ be the online size Ramsey number for forcing a red copy of $P_4$ or a blue copy of $P_n$.

Cyman–Dzido–Lapinskas–Lo conjecture. For every $n\geq 4$,

$$\tilde{r}(P_4,P_n)=\left\lceil\frac{7}{5}n\right\rceil-1.$$

The cited work had established the lower bound $7n/5-1\leq \tilde{r}(P_4,P_n)$ and an upper bound $\tilde{r}(P_4,P_n)\leq 7n/5+9$. The present paper proves the sharper upper bound $\tilde{r}(P_4,P_n)\leq\lceil 7n/5\rceil-1$ for $n\geq 10$, so the conjecture is solved for $n\geq 10$; its remaining cases are finite and are not resolved in the supplied text.