Wimbledon Common. Advice please

Can anybody advice wheteher there is an East-West route through the common suitable for road tyres?
I'd like to try a new commute route and it would be great if I could cross it. People tells me is not great for thin tyres but I'd like to know for sure and I am sure there are a few people who know it well in here.

I thought there was one, to get from Parkside, behind the Windmill, to Richmond Park, but either my navigation skills are rubbish, or the reality is that there are horse and mud paths that have to be encountered as well as tarmac sections.

Therefore, cx/mountain bike would be fine, but finest tubs are going to take a kicking.