Compact homogeneous spaces that admit a self map of degree >1

It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of such map is $>1$ (in fact I did not check it).

Question. Let $G$ be a compact Lie group and $H$ its proper connected Lie subgroup. Is it true that $G/H$ admits self-maps of degree $>1$? If not, is there some classification of such quotients and that admit / don't admit a self-map with $deg>1$?

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If $X$ is a simply connected compact manifold, then one sufficient condition for the existence of maps $X\to X$ of any sufficiently divisible degree is formality: there is a commutative differential graded algebra (cdga) $A$ and cdga maps $A\to H^*(X,\mathbb{R}),A\to \Omega^*(X)$ (the de Rham complex) that both induce isomorphisms in cohomology, see Sullivan, Infinitesimal computations in topology, \S 12, in particular, Theorem 12.2.