I proved the following (in)equalities, solving my open problem which stood for a few months:

$latex \lvert \mathbb{R} \rvert_{>} \sqsubset \lvert \mathbb{R} \rvert_{\geq} \sqcap \mathord{>}$

$latex \lvert \mathbb{R} \rvert_{>} = \lvert \mathbb{R} \rvert_{>} \sqcap \mathord{>}$

The proof is currently available in the section “Some inequalities” of this PDF file.

Note that earlier I put online some erroneous proof related to this.