Can we implement a reduced-error transmission over a channel with error frequency having $\liminf<\frac{1}{2}$ and $\limsup>\frac{1}{2}$?

We know that if a channel with error flips (in the limit) strictly less than 1/2 of the bits, we can recover asymptotically error-free transmission with an appropriate code. By bit-inversion, this applies if, in the limit, the channel flips strictly more than 1/2 of the bits.

These arguments don't actually deal with the limit, though... they simply require $$\limsup_{n\to\infty}{\frac{E\cap\left[0,n\right)}{n}}<\frac{1}{2}\text{ or }\liminf_{n\to\infty}{\frac{E\cap\left[0,n\right)}{n}}>\frac{1}{2},$$ taking $E$ to be the set of locations of the flipped bits. That is, we recover asymptotically error-free transmission if $E$ has upper density $<\frac{1}{2}$ or lower density $>\frac{1}{2}$.

Of course, there are channels with error density $\frac{1}{2}$ with capacity 0, so there's not always a useful code.

This leaves a gap I find interesting. Can we say anything about channels with error having lower density $<\frac{1}{2}$ and upper density $>\frac{1}{2}$? In particular, can we reduce the upper density by use of a (possibly standard) error-correcting code?

For the sake of argument, suppose we have a binary error channel that introduces errors with upper density $\frac{3}{4}$ and lower density $\frac{1}{4}$. Can we use this to implement a transmission with reduced upper density of error? (As a concrete goal - can we get the upper density of error below $\frac{5}{8}$?)

I suspect we can't guarantee asymptotically error-free transmission, but if it's possible, such a scheme would obviously answer my question. In any case, I have no objection to a code that sacrifices an increase in lower density of error to achieve a decrease in upper density of error.