No, that is not what it said.

The summary says "higher BMI accounted for a substantial proportion of these increased risks suggesting that residual confounding or mediation by adiposity might account for some of these remaining associations"

That means that a substantial proportion of the increased risk is explained by a higher BMI. Unfortunately the summary doesn't say how much "substantial" is.

To be specific, the regression models they use assume no measurement error in predictors. In this case BMI isn't even a direct measure of adiposity, so it's kind of obvious, but even if they had a direct measure of body fat the issue could remain because error in measurement of predictors would attenuate the ability of this covariate to eliminate confounding.

After correcting for BMI, but BMI is an imperfect proxy for adiposity: "...residual confounding or mediation by adiposity might account for some of these remaining associations"