Hamiltonian displacement conjecture for Vianna's tori and real projective planes

Let $T_{a,b,c}\subset\mathbb{C}P^2$ be Vianna's torus associated with a Markov triple $(a,b,c)$, let $\mathbb{R}P^2\subset\mathbb{C}P^2$ be the real projective plane, and let $\phi$ range over Hamiltonian diffeomorphisms of $\mathbb{C}P^2$. A displacement conjecture. There exists a Hamiltonian diffeomorphism $\phi$ such that

$$T_{a,b,c}\cap\phi(\mathbb{R}P^2)=\emptyset$$

if and only if $(a,b,c)=(2,b,c)$. This extends the known non-intersection examples, including $T_{1,1,2}$ and triples having an element equal to $2$, while the Clifford torus intersects every Hamiltonian image of $\mathbb{R}P^2$.