The funding status of big science collaborations is always a bit opaque, as there are a lot of individual actors with independent funding streams, and there's always a lot of R&D, the cost of which you can only guess in advance.
Formally ESA has moved the LISA project into a "implementation phase", meaning they intend to build it, but no money has been "pledged" as far as I know. However, LISA is a flagship Space Science programme, and the SSPs are part of the ESA budget all member states must contribute to.
Basically there's a fixed, overall budget that is then split for each project adopted. So there is definitely money but I'm not sure there is a public statement that "this particular pile of money belongs to LISA"
My understanding is that ESA has agreed to fund their portion, but other partners haven't finished that process yet - e.g. the UK's science agency has to go to the treasury and make a case for it. Similar with NASA & the other partners involved.