The cardinality conjecture for gossamer numbers

Let $*G$ denote the gossamer numbers and let $d53f$ denote the real numbers. Cardinality conjecture. The cardinality of the gossamer numbers is larger than the cardinality of the reals.

This claim is motivated by the paper's view that gossamer numbers contain infinitely many infinitesimal and infinite elements associated with real numbers, contrary to the asserted equal cardinality of the hyperreals and reals in the current theory. The source provides no resolution.