The positive-pion $\pi^+$ is comprised of 1 valence up-quark and 1 valence anti-down-quark. Assuming isospin symmetry, the parton distribution functions (PDFs) for these two valence quarks should be identical. The valence-PDF is defined as the PDF of the valence quark minus its sea component, e.g. $u^v(x)=u(x)-\bar{u}(x)$. The fact that the $\pi^+$ has one valence $u$-quark and one valence $\bar d$-quark is encoded in the standard sum-rule:
$$\int_0^1 dx\left(u(x)-\bar{u}(x)\right)=\int_0^1 dx\, u^v(x)=1$$
Of course, the PDF depends on the factorization scale $\mu_F$, used in the renormalization of collinear divergences. The convolution of the PDFs with partonic cross sections gives physical quantities which do not depend on $\mu_F$. We commonly set $\mu_F=\mu_{UV}=Q$ where $Q$ is the momentum-transfer in the presumed scattering process, in order to cancel logarithmic corrections like $\log \left(\dfrac{\mu_F}{Q} \right)$ and $\log \left(\dfrac{\mu_{UV}}{Q} \right)$.
There are certain properties of the valence PDF that I would expect to hold for all values of $Q$. First, I would expect the relevant sum-rules to hold (e.g. the aforementioned equation)).
The second is that, I would expect the distribution to be symmetric about $x=1/2$, or at least peak there. The reason is, definitely the valence $u$-quark in the $\pi^+$ is most likely to carry half the hadron's momentum. DGLAP evolution resolves quark and gluon loops (i.e. the sea) so I don't see why it would change salient features of the valence PDF. (I must be wrong about this)
However when I consult some recent calculations for the valence PDF for the pion [source], the result is not even close to being symmetric around $x=1/2$.
Should the valence PDF of the pion be symmetric about $x=1/2$, or at least peak there, for any value of $Q$? Is there any way to understand how DGLAP is "physically" affecting the valence PDF? I just want to understand the above PDF a little better.