One way function built from AES? I'm implementing a Hash based signature algorithm and this paper recommends the use of $F(X,\mathit{salt})=\operatorname{AES}_\mathit{salt}(X)\oplus X$ as a hard one way function. The salt plays the same role here as it does in password hashing and is a fixed public value that varies across small groups of calculations.

The requirement is that $F$ be preimage resistant. (IE: given $F(X,\mathit{salt})$ attacker can't find $X$ faster than brute force) 

Cryptanalysis has been done which rules out AES in contexts where the attacker gets to find approximate solutions to similar equations or gets control of key bits (EX:as hash compression function). In all these cases the attacker has degrees of freedom to work with. Here they don't.

Is $F$ preimage resistant? Can the attacker find $X$ with siginficantly less that $2^{128}$ work?
This is the Matyas–Meyer–Oseas construction.  It was introduced by engineers at IBM in 1985, and analyzed in the ideal cipher model by Black, Rogaway, and Shrimpton in 2002 among all possible iterated compression functions $h_i = f(h_{i-1}, m_i) = E_a(b) \oplus c$ where $a$, $b$, and $c$ are message blocks $m_i$ or chaining values $h_{i-1}$.  For this case ($i=1, j=6$, or $a = h_{i-1}$, $b = m_i$, $c = m_i$), in the ideal cipher model, the adversary's probability of finding a preimage after $q$ queries lies in $[0.4 q/2^n, 2q/2^n]$.

Now, AES is not an ideal cipher—at best, it is a pseudorandom permutation family.  But it is likely that the related-key attacks of Biryukov and Khovratovich do not substantively affect the security of this scheme when the salt is not under attacker control.

(All the references you need are actually in the paper you cited already, right where it recommends using AES.  If you can't follow them because of morally reprehensible academic paywalls, Sci-Hub and LibGen are your friends.)