Largest Number Not Expressible as Sum of Fewer than 8 Cubes

Theorem
$8042$ is (probably) the largest positive integer that cannot be expressed as the sum of fewer than $8$ cubes.

Proof
$8042 = 1^3 + 4^3 + 4^3 + 10^3 + 10^3 + 10^3 + 17^3$, among many other expressions.

However:

$8042$ is conjectured to be the largest positive integer that cannot be expressed as the sum of fewer than $\bf 7$ cubes.

$\bf {454}$ is proven to be the largest positive integer that cannot be expressed as the sum of fewer than $8$ cubes.