  One of the main aims of this paper is to give a large class of strongly solid
compact quantum groups. We do this by using quantum Markov semi-groups (QMS's)
and non-commutative Riesz transforms. We introduce a property for QMS's of
central multipliers on a compact quantum group which we shall call approximate
linearity with almost commuting intertwiners. We show that this property is
stable under free products, monoidal equivalence, free wreath products and dual
quantum subgroups. Examples include in particular all the (higher dimensional)
free orthogonal easy quantum groups.
  We then show that a compact quantum group with a QMS that is approximately
linear with almost commuting intertwiners, satisfies the immediately
gradient-$\mathcal{S}_2$ condition from [Cas21] and derive strong solidity
results (following [Cas21], [OzPo10], [Pet09]). Using the non-commutative Riesz
transform we also show that these quantum groups have the Akemann-Ostrand
property; in particular the same strong solidity results follow again (now
following [Iso15b], [PoVa14]).
