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

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, reported that $30$ was the smallest positive integer that had not been so represented.

Such a representation was found in $1999$:
 * $30 = 2220422932^3 + \paren {-2218888517^3} + \paren {-283059965^3}$

As of $2019$, the smallest positive integer for which such a representation has not been found is $114$.