I’m European and I’m extremely sure that when 90% of people hear "x m^3" they don’t think of it as "1000*x litres". And the majority of people likely can’t recall on the spot that 1 l = 1 dm^3 (a unit nobody uses), even if they remember that you multiply by 1000 to go from m^3 to dm^3. In any case most people don’t do conversions like that unless someone specifically asks them to.

Volumes are in general very unintuitive to people due to the cubic scaling. It’s further confused by the fact that the widely used decilitre, centilitre, and millilitre do scale linearly, unlike cubic metres. So 1 l = 1 dm^3 but 1 dl = 100 cm^3…