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 $J(Δ)$ be the cover ideal of a simplicial tree $Δ.$ Is it true that $J(Δ)^s$ is componentwise linear for all $s \geq 1$? In this article, we give an affirmative answer to this problem.