Powers of vertex cover ideals of Simplicial Trees

In $2011$, Herzog, Hibi, and Ohsugi conjectured that if $J$ is the cover ideal of a chordal graph, then $J^s$ is componentwise linear for all $s\ge 1.$ In 2022, Hà and Tuyl considered objects more general than chordal graphs and posed the following problem: Let $J(\mathrm{\Delta })$ be the cover ideal of a simplicial tree $\mathrm{\Delta }\mathrm{.}$ Is it true that $J(\mathrm{\Delta }{)}^s$ is componentwise linear for all $s\ge 1$? In this article, we give an affirmative answer to this problem.