Let $f(n)$ be the maximum number of edges in a subgraph of $Q_n$ without a $C_4$, so that this conjecture is that $f(n)\leq (\frac{1}{2}+o(1))n2^{n-1}$.

Erd\H{o}s \cite{Er91} showed that\[f(n) \geq \left(\frac{1}{2}+\frac{c}{n}\right)n2^{n-1}\]for some constant $c>0$, and wrote it is 'perhaps not hopeless' to determine $f(n)$ exactly. Brass, Harborth, and Nienborg \cite{BHN95} improved this to\[f(n) \geq \left(\frac{1}{2}+\frac{c}{\sqrt{n}}\right)n2^{n-1}\]for some constant $c>0$.

Balogh, Hu, Lidicky, and Liu \cite{BHLL14} proved that $f(n)\leq 0.6068 n2^{n-1}$. This was improved to $\leq 0.60318 n2^{n-1}$ by Baber \cite{Ba12b}.

A similar question can be asked for other even cycles.

See also [666] and the entry in the graphs problem collection.
https://mathweb.ucsd.edu/~erdosproblems/erdos/newproblems/TuranInCube.html


References


[BHLL14] Balogh, J\'{o}zsef and Hu, Ping and Lidick\'{y}, Bernard and Liu, Hong, Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin. (2014), 75-85.

[BHN95] Brass, Peter and Harborth, Heiko and Nienborg, Hauke, On the maximum number of edges in a {$C_4$}-free subgraph of
{$Q_n$}. J. Graph Theory (1995), 17--23.

[Ba12b] R. Baber, Tur\'{a}n densities of hypercubes. arXiv:1201.3587 (2012).

[Er91] Erd\"{o}s, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.