The symmetric $(n,k,n)$ half-duplex relay-channel DMT conjecture

Consider a symmetric $(n,k,n)$ half-duplex MIMO relay channel, meaning that the source and destination each have $n$ antennas and the relay has $k$ antennas. Let $d^*(r)$ denote its optimal diversity order at multiplexing gain $r$, and let $d_U^s(r)$ be the upper bound defined by the piecewise expression in the source, with $p=\min\{k,n\}$. The symmetric relay-channel DMT conjecture. For a symmetric $(n,k,n)$ relay channel,

$$d^*(r)=d_U^s(r).$$

Thus the stated upper bound is tight. The equality was found by evaluating the exact DMT for several values of $n$ and $k$; no proof of tightness is supplied, so the conjecture remains open.