Given that:
Light of higher frequency carries more energy
Time dilates in the presence of energy, and moreso in the presence of more energy
Motion in a more time-dilated space appears slower to a distant observer in less time-dilated space
Does light then appear to a distant observer to travel slower at higher frequencies than at lower ones?
This question seems similar to Do all frequencies of light have the same speed? but that question lacks the details of interest this one has.
No. So far all experimental evidence is that all frequencies of light travel at the same speed. This is in accordance with the prediction of relativity.
Moreover, in alternative theories like doubly special relativity, higher frequencies are actually expected to travel slightly faster not slower.
EDIT: to clarify some points that came up in the comments:
Spacetime curvature is invariant, the same in all frames, and arises from the stress-energy-momentum tensor. How light contributes to that tensor is a subtle and tricky issue, because light's energy (and momentum) is entirely frame dependent, not invariant. So for example two light beams moving parallel to each other do not attract one another according to general relativity (see Tolman, Ehrenfest, and Podolsky "On the Gravitational Field Produced by Light"). Roughly speaking the momentum terms in the tensor have the opposite sign to the energy term, and hence cancel it out.
That doesn't mean light has no effect on spacetime curvature; for example, two light beams moving in opposite directions do have an invariant energy and hence curve spacetime (and attract one another, per Tolman et. al.) It just means that saying "light has energy and therefore curves spacetime" is too simplistic.
That's not how gravitational time dilation acts on light. It would still travel along null geodesics, but its frequency would be blueshifted towards higher frequencies. But this would in turn mean more spacetime curvature, meaning more blueshift, and we have a feedback loop going which not only violates local energy conservation but also would mean that all light will turn into a black hole, given enough time. This goes against observations, and is therefore very unlikely to be a real effect.
The 1972 paper in Physical Review D (Frequency Dependence of the Speed of Light in Space, Z. Bay and J. A. White, Physical Review D 5(4) 796-799 (1972)) analyzes a variety of data from astronomy.
Their analysis limits
the dispersion in the microwave, infrared, visible, and ultraviolet regions of the spectrum to less than one part in $10^{20}$.
No. Time has stopped for the light, whereas the light arrives at the observeer at the same speed of light irrespective of its frequency or energy.
