Smallest Positive Integer not of form +-4 mod 9 not representable as Sum of Three Cubes

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Conjecture

It is possible to express every positive integer, not of the form $\pm 4 \pmod 9$, as the sum of cubes of $3$ integers.


Progress

In $1997$, in his Curious and Interesting Numbers, 2nd ed., David Wells reported that $30$ was the smallest positive integer that had not been so represented.

Such a representation was found in $1999$:

$30 = 2220422932^3 + \left({- 2218888517^3}\right) + \left({- 283059965^3}\right)$

As of $2017$, the smallest positive integer for which such a representation has not been found is $33$.


Sources