The real Hilbert space factorization property conjecture

Let $X$ and $Y$ be real Banach spaces. The pair $(X,Y)$ has the Hilbert space factorization property (HFP) when every bounded operator $\phi:X\to Y$ satisfies $\gamma_2(\phi)=\|\phi\|$. Real HFP conjecture. A pair $(X,Y)$ of real Banach spaces has the HFP if and only if $X$ or $Y$ is a Hilbert space.

The complex case has an additional nontrivial example, while the conjecture asserts that over the reals only pairs involving a Hilbert space have this property.