@LeakyNun I'm going to prove the thing with the endomorphism ring. Let $\phi \in \operatorname{End}(\Bbb Q/\Bbb Z)$.
Then for any $n \in \Bbb N$, we can look at the image $\phi(\frac{1}{n})$. Since that will be $n$-torsion again, and every $n$-torsion element can be uniquely written as $\frac{a}{n}$ with $a \in \Bbb Z/n\Bbb Z$, we get an element $a_n \in \Bbb Z/n\Bbb Z$. We can do this for every $n$. If $kd=n$, then $\frac{d}{n}=\frac{1}{k}$, so this gives us some compatability between $a_n$ and $a_k$ if $k \mid n$. If you check the details, this is exactly the compatability you need that $…

so @Rudi_Birnbaum does this look good ?

Let $H \subset G$ be finite groups,
let $V$ be an irreducible representation of $G$, such that
when restricted to $H$, $V = \oplus V_i$.

Conjecture 1
If $W$ is an irreducible representation of $G$ that doesn't branch when restricted to $H$,
then the number of occurences of $W$ in the decomposition of $V \otimes V$ into $G$-irreducibles
is less than or equal to the number of occurences of $W$ in the decompositions of $V_i \otimes V_j$ with $i \neq j$ into $H$-irreducibles.