 Solution

Near a red giant's compact core, intense nuclear burning in a small region can produce a large $L_r/m_r$ and a steep required radiative gradient. Where this exceeds the adiabatic gradient, convection transports the released energy and mixes fuel. Core helium burning is a clear example: the strongly temperature-sensitive Triple-alpha process commonly produces a convective burning core; shell flashes can also make convective regions.^[ref]

The qualification matters: an inert degenerate helium core and a steady hydrogen-burning shell on the first-ascent red giant branch are not universally convective; conduction or radiation may dominate there. Red giants also commonly have deep convective envelopes. The physical condition is an excessive required radiative gradient, not proximity to the centre alone.