Not an answer - but I decided to delete a prior comment and repost as an answer, because I think it puts the 2-4-6-8 conjecture in a very different light than considered so far, hopefully leading to some of the experts providing deeper insights and perhaps even an answer. So the stunning fact (to me) is this:

8 numbers up to $10^{10}$, besides $0$, were found to have a unique representation; they are $23343989$, $39866594$, $54847142$, $394239767$, $1769927927$, $2321530979$, $5022744494$ and $7969623044$.

All of them are $\equiv 20\ (\textrm{mod}\ 33)$!

This I think points to a deeper arithmetic nature to the conjecture than it simply (likely) being true for compelling "probabilistic" reasons.

Not an answer - but I decided to delete a prior comment and repost as an answer, because I think it puts the 2-4-6-8 conjecture in a different light than considered so far, hopefully leading to some of the experts providing deeper insights and perhaps even a solution. So the stunning fact (to me) is this:

10 numbers up to $5\times 10^{10}$, besides $0$, were found to have a unique representation; they are $23343989$, $39866594$, $54847142$, $394239767$, $1769927927$, $2321530979$, $5022744494$, $7969623044$, $13295525747$, $14076782201$.

All of them are $\equiv 20\ (\textrm{mod}\ 33)$!

This I think points to a deeper arithmetic nature to the conjecture than it simply (likely) being true for "probabilistic" reasons.

UPDATE. I was wrong. As pointed out in the comments by StefanKohl and user21820, the anomaly described above can be fully explained by the fact that the $\equiv 20\ (\textrm{mod}\ 33)$ residue class appears with only 48.5% of the expected $\frac{1}{33}$ frequency. The next least frequent residue class is $9$, with 64% of the expected frequency. Since in the ranges considered, every number has more than 50 hits on average, that already makes the single hit probability millions of times higher for class $20$ than class $9$. I verified that also double, triple, etc. hits are quickly dominated by the $20$ residue class, followed far behind by the $9$.

Small upshot: a counterexample is most likely going to be $\equiv 20\ (\textrm{mod}\ 33)$ , and there are low-memory search algorithms that would be faster when restricted to that case. I'll post in the comments if/when I pursue that.

Not an answer but I decided delete a prior comment and repost as an answer, because I think it puts the 2-4-6-8 conjecture in a very different light than was considered so far, and thus maybe some of the experts will provide deeper insights, possibly leading towards an answer.

Getting to the point, the stunning fact (to me) is this:

8 numbers up to $10^{10}$, besides $0$, were found to have a unique representation; they are $23343989$, $39866594$, $54847142$, $394239767$, $1769927927$, $2321530979$, $5022744494$ and $7969623044$. All of them are $\equiv 20\ (\textrm{mod}\ 33)$!

This I think points to a deeper arithmetic nature to the conjecture, than it simply (likely) being true for compelling "probabilistic" reasons!

Not an answer - but I decided to delete a prior comment and repost as an answer, because I think it puts the 2-4-6-8 conjecture in a very different light than considered so far, hopefully leading to some of the experts providing deeper insights and perhaps even an answer. So the stunning fact (to me) is this:

8 numbers up to $10^{10}$, besides $0$, were found to have a unique representation; they are $23343989$, $39866594$, $54847142$, $394239767$, $1769927927$, $2321530979$, $5022744494$ and $7969623044$. 

All of them are $\equiv 20\ (\textrm{mod}\ 33)$!

This I think points to a deeper arithmetic nature to the conjecture than it simply (likely) being true for compelling "probabilistic" reasons.