Winnie-the-Pooh conjecture for orthogonal decompositions of $sl(n)$

Let $L=sl(n)$ be the simple Lie algebra over an algebraically closed field of characteristic zero, with Killing form $K$. An orthogonal decomposition is a direct-sum decomposition

$$L=\bigoplus_{i=1}^{h+1}H_i$$

of $L$ into Cartan subalgebras such that $K(h_i,h_j)=0$ for all $h_i\in H_i$, $h_j\in H_j$ whenever $i\ne j$.

Winnie-the-Pooh conjecture. The Lie algebra $sl(n)$ has an orthogonal decomposition if and only if

$$n=p^m$$

for a prime number $p$.

This conjecture characterizes the dimensions in which orthogonal decompositions of $sl(n)$ are expected to exist. The source attributes it to A. I. Kostrikin, I. A. Kostrikin and V. A. Ufnarovsky; no resolution is supplied here.