Dichotomy of the spectra of directed graphs
The answer to “alternatively, can a directed graph have an eigenvalue with an exponentially small imaginary component” is YES (though I don’t understand what is “alternative” about this statement, as it does not in any way disprove the conjecture).

As I already wrote in a comment, it is a not very difficult exercise to show that if $f\in\mathbb Z[x]$ is a monic polynomial, there exists a directed graph $G$ on $O(\deg(f)+\|f\|_1)$ vertices whose eigenvalues include all roots of $f$. This implies the statement above modulo the fact that such polynomials may have roots with exponentially small imaginary part, which I assumed to be easy to find in the literature on minimal root separation. However, I realized that such bounds are actually not as readily found as I expected, so I decided to write it as a proper answer for the record.

Several examples of polynomials with exponentially small root separation are listed by Schönhage [1], in particular the family of polynomials
$$x^n-2(cx-1)^2\qquad(n\ge3,c\ge2)$$
attributed to Mignotte [2] (which I cannot verify as I have no access to it at the moment). Now, these polynomials have each a pair of real roots near $1/c$ in distance $&lt;2/c^{1+n/2}$, whereas we need a pair of complex roots. This is, however, easily accomplished by modifying the polynomial slightly: let
$$f(x)=x^n\mathbin{\color{red}+}(2x-1)^2=x^n+4x^2-4x+1.$$
Clearly, this polynomial has no positive real root (and no negative real root either if $n$ is even). Moreover, it is easy to show that it still has a pair of (necessarily non-real) roots in exponentially small distance to $1/2$; if I didn’t mess up the computation, these roots are approximately
$$z_\pm=\frac12\pm i2^{-1-\frac n2}+O(n2^{-n}).$$
Now, $f(x)$ can be written as the determinant of e.g. the $n\times n$ matrix
$$\begin{pmatrix}
x&amp;-1\\
&amp;x&amp;-1\\
&amp;&amp;x&amp;-1\\
&amp;&amp;&amp;\ddots\\
&amp;&amp;&amp;&amp;x&amp;-1\\
1&amp;-4&amp;4&amp;&amp;&amp;x
\end{pmatrix}$$
and therefore as the characteristic polynomial of the adjacency matrix of the weighted directed graph $G_0$ on $n$ vertices $\{0,\dots,n-1\}$ with edges $i\to i+1$ of weight $1$ for $i=0,\dots,n-2$; $n-1\to0$ of weight $-1$; $n-1\to1$ of weight $4$; and $n-1\to2$ of weight $-4$. The eigenvalues of $G_0$ are thus exactly the roots of $f$, including $z_\pm$.

Finally, the eigenvalues of $G_0$ are included among eigenvalues of the unweighted directed graph $G_1$ on $2n+6$ vertices
$$0_+,0_-,\dots,(n-2)_+,(n-2)_-,(n-1)_+^0,\dots,(n-1)_+^3,(n-1)_-^0,\dots,(n-1)_-^3$$
with edges $i_+\to(i+1)_+$ and $i_-\to(i+1)_-$ for $i=0,\dots,n-3$; $(n-2)_+\to(n-1)_+^j$ and $(n-2)_-\to(n-1)_-^j$ for $j=0,\dots,3$; $(n-1)_+^0\to0_-$, $(n-1)_-^0\to0_+$; and $(n-1)_+^j\to1_+$, $(n-1)_+^j\to2_-$, $(n-1)_-^j\to1_-$, $(n-1)_-^j\to2_+$ for $j=0,\dots,3$.

References:

[1] A. Schönhage, Polynomial root separation examples, Journal of Symbolic Computation 41 (2006), no. 10, pp. 1080–1090, doi:10.1016/j.jsc.2006.06.003.

[2] M. Mignotte, Some useful bounds, in: Buchberger, Collins, Loos (eds.), Computer Algebra: Symbolic and Algebraic Computation, 2nd ed., Springer-Verlag, 1983, pp. 259­–263, doi:10.1007/978-3-7091-7551-4_16.