Extremal spectral result of outerplanar graphs without $P_{3\cdot l}$

A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. Let $\mathrm{spex}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs. For integers $t\geq1$ and $l\geq2$, let $P_{t\cdot l}$ be the starlike tree with $t$ branches of length $l-1$. For sufficiently l