As counterintuitive as it may seem there are no ways in holacracy to "get the work done", meaning there's basically no way to oblige anyone to start doing something (there are although ways to restrict the way things are done) . All you can do is to make clear what is expected from role fillers and rely on the fact that the way they prioritize their work is the best fit to serve the organization purpose (when in doubts the can always rely on the LL for prioritizazion of their work).
In this specific instance it feels like everything else the Lead Link had to do (either in his LL role or any other role he might be filling) was currently perceived by him as something with higher priority with respect to that newly created accountability on the new role:
1) Had the role been assigned to someone else, that person might have prioritized differently --> Did anyone self candidate for that role before the LL self assigned? Did the Lead Link try to assign the role to someone else?
2) If the Lead Link is spending a lot of time in his LL role and very little in other roles you might have a first sign of warning, as he is probably still doing a lot of stuff implicitly --> Lead Link might need some coaching (he might be controlling stuff he should not be controlling, creating some bottlenecks and overheads the removal of which could create space for fulfilling the new role we are talking about)
3) There is a chance your org is understaffed with respect to the desired tradeoff between quantity, quality and speed of the things it has to do to fulfil its purpose