Let us recall this fact. Let $G$ be a semisimple algebraic group over $mathbb C$ and let $V,V’$ be two irreducible $G$-representations. We denote by $X,X’$ the unique closed $G$-orbits contained in $mathbb P V, mathbb P V’$ respectively. We know that if
mathbb P V supset X cong X’ subset mathbb P V’
as projective $G$-varieties, then $mathbb PV cong mathbb PV’$ as projective spaces. In particular, $dim V=dim V’$.
I want to understand the inverse direction: if I have two irreducible $G$-representations $W,W’$ of the same dimension, should I conclude that the closed $G$-orbits $Y subset mathbb P W, Y’ subset mathbb P W’$ are isomorphic as projective $G$-varieties?