To avoid support issues with webp (see https://caniuse.com/webp some browser like IE, which I suspect might be common in public services, do not support webp at all), you could simply resize your images to be much smaller and keep the jpg format. The current file is so big that you'll probably get 99% of the win just with that.
If you have a bit more time, you can probably look into media queries to serve the most appropriate image size depending on the resolution of the client. Alaveteli might actually do that for you already, I'm not sure.
Anyway, long story short, start by resizing the images you have down to maybe 1280px width or something, without changing the filenames or anything. I suspect that pagespeed's advice will already change (for comparison, your image weighs 9.8MB, madada's 186kB, and wdtk 60kB, both jpg). I just ran the checks on our front page, and for us the main advice is "compress text", image size checks are all green (although we could shrink the image a little bit more apparently)