Rank monotonicity conjecture for induced representations of Clifford algebras

Let $C_f$ and $C_g$ be the Clifford algebras associated to forms $f$ and $g$, respectively, and let a representation of $C_g$ induce a representation of $C_f$. Write the ranks of these representations for their dimensions as finite linearizations.

Rank monotonicity conjecture. The obtained representation of $C_f$ has rank no less than the rank of the initial representation of $C_g$.

The preceding example motivates this assertion, while the supplied text gives no proof or counterexample. Its later use is conditional on the conjecture being true.