It's all rotten, and I suspect the ends of the slate battens might be similar. Not only that, but the "blockiness" of the rot suggest dry rot rather than wet rot. Either way, it's all got to come off and something new put in place. It really needs scaffolding, stripping and sorting with new. Anything else is a bodge. This is a good retro-fit dry verge system for slates that looks very neat https://www.kytun.com/product/slate-dry-verge-system-retro-fit/ If the batten ends are soft it will help retain the slate edge.
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