The maximal-independent-set conjecture for graphs with perfect matchings

Let $\Gamma$ be a graph on $n$ vertices containing a perfect matching. A maximal independent set is an independent vertex set that is not properly contained in any larger independent set. The perfect-matching maximal-independent-set conjecture. $\Gamma$ contains at most

$$2^{n/2}$$

maximal independent sets. This is posed as a conjecture about extremal enumeration in graphs with perfect matchings; the source gives no resolution, so the asserted bound remains open.