On extension of overconvergent log isocrystals on log smooth varieties

By works of Kedlaya and Shiho, it is known that, for a smooth variety $\overline{X}$ over a field of positive characteristic and its simple normal crossing divisor $Z$, an overconvergent isocrystal on the compliment of $Z$ satisfying a certain monodromy condition can be extended to a convergent log isocrystal on $\left(\overline{X}, \mathcal{M}_Z\right)$, where $\mathcal{M}_Z$ is the log structure associated to $Z$. We prove a generalization of this result: for a log smooth variety $\left(\overline{X},\mathcal{M}\right)$ satisfying some conditions, an overconvergent log isocrystal on the trivial locus of a direct summand of $\mathcal{M}$ satisfying a certain monodromy condition can be extended to a convergent log isocrystal on $\left(\overline{X}, \mathcal{M}\right)$.